8th Grade - Gateway 2

The materials reviewed for CK-12 Interactive Middle School Math 8 for CCSS meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade level.

Chapters 1 and 2 have multiple opportunities for students to independently develop conceptual understanding of congruence and similarity using physical models, transparencies, or geometry software (8.G.A) through the use of Interactives. Examples include:

• In Lesson 1.5, first Interactive, students explore rotations to find shapes that are congruent. The student directions state, “Using the Interactive below, match any shapes that are congruent. Then answer the questions below it.” (8.G.2)

• In Lesson 2.3, Activity 2, students develop understanding of finding angle measurements based on the side lengths of the triangle. The Interactive has students manually adjust the triangle to find the angle measurements. The student directions state, “Use the Interactive below to find the angles of a triangle with the ratio 2:3:4.” (8.G.5)

• In Lesson 2.5, Activity 1, students develop understanding of similarity by identifying relationships between pairs of shapes. The student directions state, “Use the Interactive below to determine whether the two shapes are congruent, similar or neither.” (8.G.4) 

Chapter 7 has multiple opportunities for students to work independently to build conceptual understanding of defining, evaluating, and comparing functions and using functions to model relationships between quantities (8.F.A,B) through the use of Interactives. Examples include:

• In Lesson 7.1, Activity 2, students use an Interactive to explore different numbers and operations when creating functions, which helps students understand what a function is. The student directions state, “All functions need a rule for determining the outputs for corresponding inputs. This rule can be complicated or as simple as adding one to the input. Use the Interactive below to create your own function.” (8.F.1)

• In Lesson 7.3, Activity 2, students work in the Interactive by moving points on a coordinate plane and identify if a function is created. This develops understanding of what a function is and what the graph of a function looks like. The way to identify a function is stated as, “To determine whether a relation is a function, you need to check whether one input value leads to two different output values. If one input value does lead to two different output values, you will be able to tell visually because the two points will line up vertically. You can often find this using what is called ‘The Vertical Line Test.’” (8.F.1)

• In Lesson 7.6, Activity 2, students write a function for a given graph, which develops understanding of analyzing graphs for functions. This is introduced to the students as, “Construct a function to represent the following situation: An app developer is looking at a graph which shows active users as a function of time. This means that the input value is the time and the output value is the number of people using the app at that time. The time 0 hours represents 12:00 AM EST. The graph shows that from 0 hours to 7 hours, the number of users stays constant at 600. From 7 hours to 12 hours, the number of users increases to 2,000. The number of users stays constant at 2,000 until 20 hours at which point the number of users decreases to 800 at 24 hours. Construct this function below.” (8.F.5)

The instructional materials reviewed for CK-12 Interactive Middle School Math 8 for CCSS meet expectations for attending to those standards that set an expectation of procedural skill and fluency. The instructional materials develop procedural skill and fluency and provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level, especially where called for by the standards (8.EE.7,8b).

In Chapter 3, the materials develop and students independently demonstrate procedural skill in solving linear equations in one variable (8.EE.7). Examples include:

• In Lesson 3.5, Review Questions, students demonstrate procedural skill in solving linear equations with a single variable on each side of the equation. Some examples include, “1. Solve: 8(2k-1)=13(2k+4); 2. Solve for y: 3(4y-2)=2(2y+5); and 5. Solve for x: 10x-4=2x+60.”

• In Lesson 3.8, students determine the number of solutions for an equation. Activity 1: Accounting For All Possibilities Continued explains how an equation can have an infinite number of solutions. The materials state, “$$.75(x+23) = .75x + 17.25$$; .75x + 17.25 = .75x + 17.25; 17.25 = 17.25. Any value for x would work so there are an infinite number of solutions.” Activity 2 contains No Solutions: “$$x = x - 1$$; if x = 0 then 0 = 0 - 1; 0 = -1. Since no value for x makes the statement true, there is no solution.” The practice page in the teacher edition provides independent practice. Examples include, “1. Does the following equation have no solution, infinite solutions, or exactly one solution? \frac{1}{3}(6k + 12) = 2k - 2 and 4. Does the following equation have no solution, infinite solutions, or one solution? 8(t - 1) = 2(4t - 5).”

In Chapter 5, the materials develop and students independently demonstrate procedural skill in solving systems of two linear equations in two variables algebraically and estimating solutions by graphing the equations (8.EE.8b). Examples include:

• In Lesson 5.2, the Interactives provide opportunities to input equations, and the Inline Questions help to direct an analysis of the graphs. For example, in Activity 1: Changing the Game, students input the equations y = 1.08x + 1.07 and y = 1.05x + 12.6 to determine if a basketball player should take a 2 point or a 3 point shot. Activity 3: Number of Solutions uses the Interactive to determine the amount of solutions, one, none or infinite, a problem may have. For example, Inline Question 3 states, “Look at each of the following as a second equation in a system with an equation 8x + 6y = 14. Decide if each system has 1 solution, no solution or infinite solutions.” Students develop procedural skill through practice questions at the end of the lesson or from the teacher edition, including, “4. Solve the system by graphing: y^2 - x - 44 = 0 and x - 2y = 4. or 2. Find the point of intersection of the graphs of the equations y = -x and y = x - 2,” respectively. (8.EE.8)

• In Lesson 5.3, Review Questions, students solve multiple systems of equations independently using substitution. Some examples include, “1. Solve the following system of equations by substitution: x+2y-1=0, 3x-y-17=0;  4. Solve the following system using the substitution method. 2x + 3y = 5, 5x + 7y = 8; and 5. Solve the following system using the substitution method. 2x-5y=21, x=-6y+2.” (8.EE.8b)

• In Lesson 5.4, Review Questions, students solve multiple systems of equations independently using elimination. Some examples include, “3. Solve the following system of equations by elimination. Express the solution as an ordered pair (x, y). 4x=-14-6y; -5x-6y=22 and 4. Solve the system using elimination. 3y-4x=-33, -5x-3y=40.5.” (8.EE.8b)

The materials reviewed for CK-12 Interactive Middle School Math 8 for CCSS meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. Examples include:

• In Lesson 4.2, students develop their conceptual understanding of graphing proportional relationships. In Activity 2, students “Use the Interactive to populate the table, determine the relationship between weight and medicine dosage, and express that relationship as an equation.” (8.EE.5)

• In Lesson 5.1, Activity 1 Interactive states, “There are two water bottles that each hold 16.9 fluid ounces. Bottle A is slowly being filled with water at an average rate of 0.8 fluid ounces per second. Bottle B is full and has a small hole poked in it and is leaking water at an average rate of 0.5 fluid ounces per second. If Bottle B starts to drain at the same time that Bottle A begins to be filled, at what time will Bottle A and Bottle B have the same amount of water?” Through the Interactive, students apply systems of equations to the real-world context. (8.EE.8b)

• In Lesson 3.4, Activity 3, students develop procedural skill in solving 1-variable, linear equations. For example, Practice Problem 2 states, “Solve the equation: -46 = -4(3s + 4) - 6.” (8.EE.7)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• In Lesson 2.5, Activity 1 Interactive, students develop conceptual understanding of similarity, congruence, and dilations. The materials state, “Recall that when an image is dilated, this causes the side lengths to change, but the angles remain the same. Two shapes that can be produced by dilating one to obtain the other are called similar shapes. Use the Interactive below to determine whether the two shapes are congruent, similar, or neither.” In Activity 3, students apply their understanding of similarity and triangles in a real-world context. The materials state, “Thales knew that he had constructed similar triangles. Once the triangles were constructed, Thales used a proportion to compare the sides of one triangle to the corresponding sides of the other triangle to find the distance of the ship from the shore. How far was the ship from the shore in the picture below?”

• In Lesson 4.1, Activity 3, students develop conceptual understanding of graphing proportional relationships and the constant of proportionality. The materials state, “The equation for a proportional relationship is y=kx where x and y are the related quantities, and k is the constant of proportionality. Use the Interactive below to graph the relationship between minutes and the number of beats based on the equation.” A Supplemental Question states, “All of the graphs have been straight lines, do you think it is a coincidence? Why?” In the Practice problems, students develop procedural skill in solving direct variation equations.

The materials reviewed for CK-12 Interactive Middle School Math 8 for CCSS meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The materials intentionally identify and develop MP1 in connection with grade-level content by providing opportunities for the students to make sense of problems and persevere in solving them. Examples include: 

• In Lesson 1.5, Defining Congruence, Students develop and test a strategy for mapping lines through transformations.  This encourages them to make sense of the problems and persevere in finding a solution. In Activity 1: Mapping Shapes, the students are asked to generalize a strategy for mapping one shape onto another using a series of rigid transformations in as few moves as possible. The Teacher Notes state, “Rather than telling the students that they are correct or incorrect, encourage the students to come up with more creative ways to improve on their own strategies and the strategies of their classmates.”  This further encourages problem solving and perseverance. (8.G.2)

• In Lesson 5.3, Solving Systems of Equations Using Substitution, “Students solve puzzles that visually simulate systems of equations.” In Activity 1: Understanding Substitution, students work with five interactive puzzles “to develop strategies that allow them to substitute and solve for unknown variables.” The puzzles increase in difficulty after each stage, and students will have to persevere to solve all five. (8.EE.8)   

• In Lesson 7.3, Identifying Functions, students intentionally develop MP 1 during the Warm Up as they explore solution pathways to determine how a function is broken. Warm Up: Drop Down Lists, “Explore the drop-down list below and determine the relationships between position and color.” The Discussion Question asks, “Explain the relationships between the input position and the output color. What do you notice?” Students will have to go through the drop-down list multiple times in order to answer the question. (8.F.1)

The materials intentionally identify and develop MP2 in connection with grade-level content by providing opportunities for the students to reason abstractly and quantitatively. Examples include:

• In Lesson 6.2, Linear Patterns in Scatter Plots, MP2 is intentionally developed throughout the lesson using the Interactives, Inline Questions, and Discussion Questions as they contextualize linear trends. Activity 1: Positive Trends, looks at positive trends, while Activity 2: Negative Trends, looks at negative trends.  Inline Questions from Activities 1 and 2 include the following: “Which of the following data sets would have a positive trend? Which of the following data sets would have a negative trend? Which graph has a weaker relationship, and how do you know?”  The Discussion Question asks the following: “Which graph has a stronger relationship? How do you know? If there is a relationship between engine size and efficiency, what could cause larger engines to be less efficient?” (8.SP.1)

• In Lesson 7.6, Interpreting Graphs of Functions, “students are given app usage as a function of time and asked to construct a scenario which would fit this data.” In Activity 1: Understanding a Graph in the Context, students are asked to, “use the Interactive below to examine two sample velocity vs. time graphs and break down what happened over the course of the functions.” This analysis requires students to reason quantitatively and abstractly. (8.F.5) 

• In Lesson 10.10, Finding Distance Using the Pythagorean Theorem, students reason abstractly and quantitatively as they use an Interactive to investigate finding the shortest distance between two points. Students are asked first, “1. Mark all true statements: (1) Routes A and C have equal distances. (2) Route B is shorter than route A.” And then, “2. The most optimal route would be a straight line from school to home.  How long is the optimal route? Round your answer to the nearest tenth.” (8.G.8)

The materials reviewed for CK-12 Interactive Middle School Math 8 for CCSS meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials intentionally identify and develop MP3 in connection with grade-level content by providing opportunities for the students to construct viable arguments and critique the reasoning of others. Examples include:

• In Lesson 1.11, Sequences of Transformation on the Coordinate Plane, Standards for Mathematical Practice: “MP3: In activity 1, the students are asked if order matters when performing a sequence of transformation on a coordinate plane. The students must cite evidence to support their argument.” Activity 1: Identifying Sequences of Rigid Motions, Discussion Question states, “Jessie wants to reflect and translate a point. She believes that the order in which she does so does not matter. Does it matter if she translates the point first and then reflects it compared to if she reflects it first and then translates it? As long as the translation and reflection are the same, will the point end up in the same place?” (8.G.2 & 8.G.3 )

• In Lesson 4.7, Investigating Horizontal and Vertical Lines, Standards for Mathematical Practice are as follows:  “MP3: In Activities 1 & 2, the students are asked to make conjectures about horizontal and vertical lines and their slopes through discussion with classmates.” Warm-up: What Do Horizontal and Vertical Lines Mean? Discussion Question #3: “Discuss with a classmate how we can use our knowledge that the x-axis and y-axis are perpendicular to each other to prove that horizontal and vertical lines are perpendicular to each other? Do you agree with your classmate, why or why not?” Activity 1: Horizontal Lines, Discussion Question #2 “Discuss with a classmate what would the slope of all horizontal line equations be 0? Why or why not?” Activity 2: Vertical Lines, Discussion Question #2 “Discuss with a class would the slope of all vertical line equations be undefined? Do you agree with your classmate? Why or why not?” (8,EE.B)

• In Lesson 6.4, Fitting a Line to Data, Standards for Mathematical Practice are as follows: “MP3: In Activity 1, the students form an argument over the factors that separate a good line of fit from the best line of fit and their necessity.” Activity 1: Line of Best Fit, Discussion Question: “Manny says, "It doesn't matter if a line of best fit is perfect because it's not going to match the data anyway. As long as you are close it's fine." Do you agree or disagree with Manny? Support your argument with evidence.” (8.SP.2 & 8.SP.1)

• In Lesson 7.1, Introduction to Functions, Standards for Mathematical Practice are as follows: “MP3: In Activity 1, the students form arguments about the pros and cons of each function representation.” Activity 1: Visualizations of Relations and Functions, Discussion Question: “Grace claims that since each method displays the same function, it doesn't matter which you choose. Do you agree or disagree with Grace? When might one method for displaying a function be better than another?” (8.F.4)

The materials reviews for CK-12 Interactive Middle School Math 8 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students,  in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The materials intentionally develop MP6 through providing instruction on communicating mathematical thinking using words, diagrams, and symbols. Examples include:

• In Lesson 2.4, Properties of Dilations, students are introduced to scale factor and dilations. In the Warm Up, students are reminded, “Transformations change the location or orientation of an image but not the shape. Rigid motions are transformations that move an image, but do not change the size. The only transformation that is not a rigid motion is a dilation. A dilation is a transformation that changes the size of a figure.” In Activity 1, students read directions for the interactive and are also given important vocabulary. “To perform a dilation, you need to specify a scale factor and a center of dilation. The scale factor is the number which is used to multiply the size of the image. The center of dilation is the point from which the image is being dilated.” (8.G.3 & 8.G.4)

• In Lesson 4.4, Defining Slope, Activity 2, students connect the definition of linear relationships to the example from the Warm-up. “A linear relationship is a relationship that traces a line when plotted. As you may be able to tell by the first four letters in LINEar, the word linear means arranged in a straight line. Proportional relationships are a specific type of linear relationship where the starting amount is 0. All proportional relationships are linear because they form a straight line when graphed. However, not all linear relationships are proportional because they do not have to start at 0. In the example of the firefighter, you were dealing with a linear relationship between pressure and floor number. The pressure started at a given non-zero number and then increased by 5 psi for every floor.” ( 8.EE.5 & 8.EE.6)

• In Lesson 7.1, Introduction to Functions, MP6 is intentionally developed as the students “explore the different ways to express a function,” such as ordered pairs, in a table, in a mapping, a graph, or an equation. The Teacher Notes include how the language relating to functions will connect with students’ prior knowledge. “In this lesson, the language about input/output/functions is connected to prior learning in 6th grade relating to independent and dependent variables.” (8.F.4)

The materials use precise and accurate terminology and definitions when describing mathematics, and the materials also support students in using the terminology and definitions. There is no separate glossary in these materials, but definitions are found within the units in which the terms are used. The vocabulary words are in bold print. Examples include:

• In Lesson 3.1, Writing Equations with Variables, students attend to precision as they answer Inline Questions with immediate feedback and accuracy. An example is in Activity 2: “In the Interactive, what is the coefficient for x and y in the given expression?” In addition, the Teacher Notes stress accurate language: “Be sure to consistently use the words ‘constant, coefficient, term, expression and equations’ with students throughout the lesson so that they can identify the different parts of the equations.”. (8.EE.7)

• In Lesson 6.1, Representing Data in Scatterplots, students are given exact definitions of bivariate and quantitative data in the Warm Up: “Bivariate data means that the data comes from two variables. For example, if the temperature of a room changes over time, the two variables are the temperature and the time at which the temperature was measured. Quantitative data is data that can be measured. Examples of quantitative data are the height of a person, the speed of a car, the mass of a baseball, etc.” In Activity 1, they answer precise Inline Questions about the bivariate quantitative data in a scatter plot., such as, “Find the coordinate (9, 63). What does this coordinate mean?” (8.EE.8)

• In Lesson 10.11, Representing Rational Numbers with Decimals, Activity 1, MP6 is intentionally developed as students “label numbers to their corresponding number systems.  Additionally explore the vocabulary associated with different types of decimal values.” In Activity 3 students answer Inline Questions that require precision of language and answers, such as Question 3: “A decimal that can never be fully written because it repeats forever is considered a rational number because it can be described as a _________.  (8.NS.1)

The materials reviewed for CK-12 Interactive Middle School Math 8 for CCSS meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

The materials intentionally identify and develop MP7 in connection with grade-level content by providing opportunities for the students to look for and make use of structure. Examples include:

• In Lesson 3.4, Linear Equations and the Distributive Property, Activity 1, the students explore how an expression representing profit can be equivalent to the difference between expressions representing revenue and cost.  Using the same equation that was used in Warm-Up: Selling Shirts, students work to find out the answer to the following statement, “Patrick wants to know how many shirts he will have to sell to reach this goal ($400,000) without changing the sales price or costs.” Students manipulate the structure of the previous equation to answer this statement.Then students are asked questions about their results, “Discussion Questions, 1)”You  combined 15q and -6.5q to get 8q.  What does this 8q represent? 2) Does the answer "62,000 shirts" make sense within the context of the problem? How can you check your answer mathematically? Does it check out mathematically?” (8.EE.7)

• In Lesson 4.10, Linear Equations in Standard Form, students intentionally develop MP7 as they “explore how a linear equation in standard form can be written in slope-intercept form and vice-versa.” Throughout the lesson students are shown different ways to graph an equation, for example graph by plotting points or graph by plotting the intercepts. All the linear equations in all activities start off in standard form. In Activity 3: Graph by Converting to Slope-Intercept Form, students examine the structure of a linear equation in standard form and solve it for y using the same structure they would solve a two-step equation. (8.EE.5 & 8.EE.6)

• In Lesson 5.4, Solving Systems of Equations Using Elimination, Activity 3: Rearranging Equations, the students examine how the placement of the terms in an equation affects their ability to eliminate a variable. “Solve the system of equations using the elimination method:

  2x + 4y = 40

  x  = 4y - 22

  ”Students examine the structure of the system of equations as they follow the steps in solving it. Students also examine the structure of the system of equations when the Inline Questions asked them to consider two more system of equations, one which they have to add without structural manipulation to the equations, resulting in no variable being eliminated and one where they are asked to pick the operation that would eliminate a certain variable. (8.EE.8b)

The materials intentionally identify and develop MP8 by providing opportunities for the students to look for and express regularity in repeated reasoning. Examples include:

• In Lesson 4.1, Graphing Proportional Relationships, students intentionally develop MP8 as they “use repeated reasoning to look for shortcuts in identifying the points of a proportional relationship on a graph.” In Activity 1: Proportional Relationships, students are tasked with using an interactive graph to show Emily’s earnings as the number of hours that she babysits increases. Students are tasked with graphing at least five points in order to use repeated reasoning to answer the Inline and Discussion Questions. For example, Discussion Question 3 asks, “Did you notice a pattern while you were plotting points? What was it and how might you be able to use this pattern when plotting in the future?” (8.EE.5)

• In Lesson 7.3, Identifying Functions, students intentionally develop MP8 as they “use repeated reasoning...to derive the vertical line test.” In Activity 1: Functions vs. Relations, functions are defined, students are shown several different representations of relations that are not functions, and then they graph four functions on different graphs.  Discussion Question #3 asks, “What similarities do you notice between graphs that form functions and graphs that do not form functions?” The Teacher Notes states, “This question encourages students to use repeated reasoning to derive a version of the vertical line test. By repeatedly graphing relations that are functions and relations that are not functions, the students begin to see that a relation that is not a function can be identified by overlapping points.”(8,F.1)

• In Lesson 8.7, Finding Dimensions of Spheres: In Activity 3, the students use repeated reasoning to see how changing the radius changes the size of the sphere proportionally. “The lesson Volume of Pyramids discussed how the dimensions affected the volume of a pyramid. A sphere has only one variable in the formula, the radius.  Use the interactive below to see (how) the radius affects the volume of a sphere.” The following Inline Questions reinforce the relationship. For example, Question 2, “In the interactive, scaling the sphere by half (or 0.5) means…”  and Question 3: “In the interactive the original sphere has a radius of 2 and volume of 33.51. Scaling the original sphere by 3 will result in a sphere with a radius of 6. What will its volume be? Use the interactive to see!”  (8.G.9)