8th Grade - Gateway 1

The materials reviewed for MathLinks: Core 2nd Edition Grade 8 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Within the MathLinks: Core 2nd Edition materials, the quizzes and cumulative tests are found online in the Teacher Portal in PDF and editable Microsoft Word versions. Cumulative tests are primarily multiple-choice, while quizzes are typically short answer. Materials assess grade-level standards and do not include above-grade assessment items. Examples include:

• Unit 2, Quiz A, Problem 1, students evaluate perfect squares and make rational approximations of expressions. “Write the whole number that is equivalent to each radical expression. If not possible, write the two consecutive whole numbers that it falls between. A)\sqrt{81},  B) \sqrt{15},  C) \sqrt{49}, D) \sqrt{121}.” (8.EE.2, 8.NS.2)

• Cumulative Tests, Test 3, Problem 2, students determine equivalent expressions involving exponents. “Choose ALL expressions that are equivalent to (x^3)^4. A) x^7, B) x^{12}, C) 12x, D) x^6\cdot x^6.” (8.EE.1)

• Unit 5, Task - Parallel Line Function, Problem 1, students graph and label parallel lines and intersection points. “Lines A and B are parallel. The equation representing line A is -2y = x.  Line B passes through the point (2, 2). Graph and label lines A and B using any method.” (8.EE.6, 8.F.4)

• Unit 8, Quiz B, Problem 6, students investigate associations in bivariate data. “Destiny got a summer job. The graph on the right shows how much money she has through day 8 of summer. C) Make a prediction using the equation of how much money she would have after working 20 days.” (8.SP.A)

• Cumulative Tests, Test 9, Problem 1, students demonstrate understanding of transformations. “Choose ALL of the following statements that are true. A) Translations, rotations, and reflections are three examples of transformations. B) A transformation of the plane is a function that maps the plane to the plane. C) A transformation takes points in the plane to points in the plane. D) A rotation is a transformation that reverses orientation.” (8.G.1)

The materials reviewed for MathLinks: Core 2nd Edition Grade 8 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. 

Materials present all students with extensive work with grade-level problems. Examples include:

• Unit 4, Lesson 4.3, To School and Back Home, Problem 1, students find the output given an input and graph the ordered pairs. “Nellie walks to school each morning at a constant rate of 0.05 miles per minute, and jogs home in the afternoon at a constant rate of 0.08 miles per minute. School and home are \frac{4}{10} of a mile apart. 1) Fill in both columns in the table below and draw graphs based upon the given data.” The table provides minutes elapsed (x) from 0 to 8. To School: Miles from home (y), and To Home: Miles from home (y) with column 0 filled in. Quadrant 1 of a coordinate plane with numbering started is provided for students to draw graphs.  (8.F.1) 

• Unit 8, Lesson 8.3, Practice 6, Problem 3, students work with simultaneous linear equations in real-world problems. “A yellow hot air balloon is 750 feet above the ground and rising at a constant rate of 3 feet per second. A blue hot air balloon starts on the ground and is rising at a rate of 8 feet per second. How long with it take for the blue balloon to reach the same altitude as the yellow balloon?” (8.EE.8)

• Unit 9, Lesson 9.3, About Congruence, Problem 2, students complete transformations to understand congruence between figures. “Using the pre-image, create Image A using the following two steps, and then label the new image. Step 1: rotate the preimage clockwise 90\degree around the origin. Step 2: reflect the result about the y-axis. Why is Image A congruent to the pre-image?” (8.G.2)

Materials present opportunities for all students to meet the full intent of the standard. 

• In both the student and teacher editions, grade-level standards for each unit are listed. If the standard is only partially addressed during the unit, the remainder of the text is struck through then identified in a different unit, making it clear when the full intent has been met. For example: 8.EE.2 - “Use square root and cube root symbols to represent solutions to equations of the form x^2=p and x^3=p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that \sqrt{2} is irrational” is first addressed in Unit 2 when students evaluate square roots of perfect squares and learn that \sqrt{2} is irrational. This standard is addressed again in Unit 3, covering all parts except knowing that \sqrt{2} is irrational. And in Unit 10, students again evaluate square roots of perfect squares. Example problems for 8.EE.2 include:

• Unit 2, Lesson 2.3, Another Well-Known Irrational Number, Problem 2, “Jordan used a calculator and found the (1.4142135)^2=2.00. Does this mean that Jordan found an exact value for \sqrt{2}? Explain.”

• Unit 3, Lesson 3.3, More Exploring With Exponents and Roots, Problems 8-9, “Compute each cube root. Recall that \sqrt[3]{64}=4\cdot4\cdot4=64.  8) \sqrt[3]{8}, 9) \sqrt[3]{-8}”

• Unit 10, Lesson 10.1, Getting Started, Problem 2, “Simplify each expression below. a) \frac{\sqrt{36}}{\sqrt{9}}, b) \sqrt{\frac{36}{9}}”

The materials reviewed for MathLinks: Core 2nd Edition Grade 8 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

To determine the approximate amount of time spent on major work of the grade, materials were analyzed from three different perspectives; units, lessons, and hours. Lesson reviews, unit reviews, and assessment days are included. In addition, supporting work that connects to major work is included.

• The approximate number of units devoted to major work of the grade is 8 out of 10, which is approximately 80%.

• The approximate number of lessons devoted to major work is 24 out of 30, which is approximately 80%. 

• The approximate number of hours devoted to major work of the grade is 112 out of 140, which is approximately 80%. One hundred forty hours includes all lessons, reviews, and assessments, but it does not include time indicated for intervention, enrichment, and school obligations as those needs vary. 

A lesson-level analysis is most representative of the instructional materials, because the lessons include major work, supporting work connected to major work, and have the review and assessment embedded. Based on this analysis, approximately 80% of the instructional materials for MathLinks: Core 2nd Edition Grade 8 focus on major work of the grade.

The materials reviewed for MathLinks: Core 2nd Edition Grade 8 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. Connections between supporting and major work enhance focus on major work.

Connections between supporting and major work enhance focus on major work of the grade. Examples include:

• Unit 2, Lesson 2.1, Practice 2, Problems 6-9 connect the supporting work of 8.NS.2 (Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram…) to the major work of 8.EE.2 (Use square root and cube root symbols to represent solutions to equations of the form x^2=p and x^3=p, where p is a positive rational number…) Students approximate the square root of 20 to evaluate different scenarios. “For their house, Greg and Lauren bought a square rug with an area of 20 square feet. Explain all answers below. 6) If the dimensions of their front entry is 5 feet by 5 feet, will the rug fit? 7) Greg decides he would rather put the rug in front of the kitchen sink, which is a space 4 feet wide. Will the rug fit in that space? 8) Lauren thinks the rug will look great in the hallway, which is 4\frac{1}{2} feet wide. Will the rug fit? 9) Greg measured the hallway again, and discovered it is actually 4 feet 4 inches wide. Will the rug fit?”

• Unit 2, Lesson 2.2, Practice 4, Problem 6 connects the supporting work of 8.NS.2 (Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram…) to the major work of 8.G.7 (Apply the Pythagorean Theorem to determine unknown side lengths in right triangles). Students use Pythagorean’s Theorem with an irrational side length. “6) To get from home to work every day, Samos drives about 7 miles south on Avenue A, and then drives east on Avenue B. He knows that the straight-line distance from his home to his place of work is about 20 miles. How many miles does he drive east on Avenue B? If Samos could drive in a straight line, “as the crow flies,” about how much shorter would his daily commute be?”

• Unit 6, Lesson 6.2, Obesity Rates by State, Problems 1-3 connect the supporting work of 8.SP.3 (Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept) to the major work 8.F.5 (Describe qualitatively the functional relationship between two quantities by analyzing a graph…) Students analyze data relating obesity and exercise. “1) Estimate (draw) a line of best fit on the graph below and write its equation; 2) Explain what the slope and y-intercept represent in the context of the problem; 3) Does the scatterplot represent a function? The line of best fit?”

The materials reviewed for MathLinks: Core 2nd Edition 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.

There are connections from supporting work to supporting work and/or from major work to major work throughout the grade-level materials, when appropriate. Examples include:

• Unit 1, Lesson 1.2, Ice Cream Cones, Problem 2, connects the supporting work of 8.G.C (Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.) and supporting work of 8.NS.A (Know that there are numbers that are not rational, and approximate them by rational numbers). Students compare the volume of ice cream they would get in a sugar cone vs a cake cone. “An ice cream store has two different kinds of cones. For a single scoop, they fill the cone with ice cream and then put a dome (half sphere) of ice cream on the top. Below are the dimensions and prices for one scoop. *Remember that d=2r. Rank the amount of ice cream from least to greatest. Show formulas and substitutions.” Students are provided with a visual and the heights and diameters of the cones. For the cake cone, they are directed to consider it as two cylinders. 

• Unit 4, Lesson 4.3, To School and Back Home, Problems 1-6, connects major work of 8.EE.B (Understand the connections between proportional relationships, lines, and linear equations), with major work of both 8.F.A (Define, evaluate, and compare functions) and 8.F.B (Use functions to model relationships between quantities). Students create and analyze multiple representations of information. “Nellie walks to school each morning at a constant rate of 0.05 miles per minute, and jogs home in the afternoon at a constant rate of 0.08 miles per minute. School and home are \frac{4}{10} of a mile apart. 1) Fill in both columns in the table below and draw graphs based upon the given data. 2) Why does it make sense to draw lines for the graphs with this context? 3) Which graph is increasing? Decreasing? 4) Does either one of these situations represent a proportional relationship? Explain. 5)  For walking to school, what is the unit rate? 6) Write an equation for each situation.”

• Unit 8, Lesson 8.3, Practice 6, Problem 4, connects the major work of 8.EE.C (Analyze and solve linear equations and pairs of simultaneous linear equations.) with major work of 8.F.B (Use functions to model relationships between quantities.) Students set up a simultaneous linear equation. “A green hot air balloon was at the maximum allowable 3,000 feet above the ground and began to descend at a constant rate of 10 feet per second. At the same time, a red hot air balloon at 300 feet above the ground starts to rise at a constant rate of 5 feet per second. How long will it take for the two balloons to be at the same altitude?”

• Unit 10, Lesson 10.3, Practice 6, Problem 1, connects major work of 8.G.B (Understand and apply the Pythagorean Theorem) with 8.EE.B (Understand the connections between proportional relationships, lines, and linear equations). Students use the Pythagorean Theorem and similar triangles to find a distance when given a diagram. “Marcellus is 5 feet tall. He casts a 7-foot shadow. At the same time, the shadow of a tree is 21 feet. Approximately how tall is the tree?”

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 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.

Within the Teacher Edition, General Information, each unit provides information about relevant aspects of the content which involve the progression of mathematics. Additionally, Teacher Notes within some lessons identify when current content is building on prior learning and/or connecting to future concepts. Connections to future content and prior knowledge include:

• Unit 2, Algebra in MathLinks: Grade 7, “Algebra topics primarily appear in the CCSS-M Expressions and Equations and Ratios and Proportional Relationships domains. These areas are the focus of four units in MathLinks: Grade 7, and they extend work introduced in 6th grade.” There is a description of the development of Algebra Topics through Unit 2: Percent and Scale, Unit 3: Proportional Relationships, Unit 6: Expressions and Unit 7: Equations and Inequalities. They describe how Units 4 and 5: Rational Number operations extend the use of cups as a manipulative they learned in grade 6 to represent an unknown. Unit 6 includes equations in slope-intercept form “without formally addressing function, slope, and vertical intercept, which is done in 8th grade.”

• Unit 7, Algebra in MathLinks: Grade 7, “In Unit 6, Expressions, students use a visual context to write numerical and algebraic expressions, paving the way to greater flexibility working with variables and expressions. Equations of the y=mx+b are explored without formally addressing function, slope, and vertical intercept, which is done in 8th grade.”

• Unit 7, About the Equation-Solving Sequence, points out that as equations become more complex, students will recognize the benefits of systematic procedures. Therefore, as they are learning procedures for one-step equations in grade 6 and two-step equations in grade 7, it is reasonable to encourage solving mentally to reinforce mathematics as sense-making and value prior knowledge. The progression of lessons in the unit starts with solving mentally, then “re-introduces a more traditional balance technique from 6th grade,” and finally more complicated manipulation of rational numbers.

• Unit 8, Lesson 2, Lesson Notes S8.2a: Sketching Figures, “Students informally begin to think about whether two or more figures exactly cover one another or if sides of one are a multiple of the other. This sets students up for the 8th grade topics of congruence and similarity.”

• Unit 9, Lesson 3, Lesson Notes S9.3: Volume and Surface Area, “Students extend their work finding volumes of right rectangular prisms from 6th grade to include right prisms with other polygonal bases.” Slide 5, “Students find the surface areas of all three prisms. Because the Pythagorean Theorem is needed to compute a value of the length of KD (an 8th grade standard), its measure is given.”

Teacher Edition, Big Ideas and Connections in each unit identifies the focus concepts of the grade level and draws connections among the content specific to the current unit. “Grade 7 is organized around seven big ideas. This graphic provides a snapshot of the ideas in Unit 6 and their connections to each other.” Below the graphic, a chart listing “Prior Work” and “What’s Ahead”, and “These ideas build on past work and prepare students for the future.” Examples include: 

• Unit 6, Teacher Edition, Big Ideas and Connections, Prior Work, “Perform operations with whole numbers, fractions, and decimals. (Grades 3, 4, 5, 6); Extend the number system to include negatives. (Grade 6); Write and interpret numerical expressions. (Grades 5, 6); Solve one-step equations using non-negative numbers. (Grade 6); Explore input-output relationships. (Grade 6)”

• Unit 6, Teacher Edition, Big Ideas and Connections, What’s Ahead, “Analyze and solve linear equations in one or more variables. (Grade 8, HS); Use algebra skills to explore the world of functions. (Grade 8, HS); Use expressions and equations to create mathematical models. (Grade 8, HS)”