8th Grade - Gateway 2

The materials reviewed for MathLinks: Core 2nd Edition Grade 8 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

According to the MathLinks Program Information, in Focus, Coherence and Rigor, “Conceptual understanding, the bedrock of a MathLinks course, frequently drives the other two components of rigor. It is a MathLinks philosophy to make sure all students have the opportunity to make meaning for every concept presented, and we focus on the conceptual development of Big Ideas in depth and make them plausible through investigations, activities, and practice. This is commonly done throughout lessons in all units, oftentimes with the help of teacher Lesson Notes and Slide Decks. Opportunities for independent work within a Student Packet appear as Practice pages within lessons, in the Review section as activities, and as Spiral Reviews in subsequent units. Unit Resources on the Teacher Portal also contain problems, tasks, and projects to support conceptual development.” A table is provided that identifies multiple ‘concept development activities’ throughout the lessons. 

Materials develop conceptual understanding throughout the grade level. (Note - Lesson Notes come after the workbook page in the Teacher Edition.) For example:

• Unit 2, Lesson 2.1, Teacher Edition, Lesson Notes S2.1: A Radical Investigation, students develop conceptual understanding of rational approximations of irrational numbers (8.NS.2) as teachers guide students through drawing squares with areas that are not perfect square values. Teachers are given prompts such as, “Allow for productive struggle and see if students first realize that whole number side lengths are impossible, and then arrive at some sort of estimate. Why is a 1\times5 rectangle NOT allowed? Why is a 2\times2 square too small and a 3\times3 too big? Do you think there is a number that can be multiplied by itself to get exactly 5?” Students record whole numbers from 1-10 on tick marks below a number line and their radical equivalents above it. Then they use this information to estimate square roots by sandwiching them between two consecutive whole numbers.

• Unit 5, Lesson 5.1, The Meaning of Slope, Problems 3 - 9, students develop conceptual understanding of the meaning of slope and find the slopes of lines by counting on grids (8.EE.6). Teachers provide the line segment to analyze (from the Teacher Edition), “First circle whether a line segment has a positive or negative slope (+ or – ). Then find the slope by counting.” After finding the slopes, students compare steepness, “11) Which value is greater, the slope of line NK or the slope of line PM? Which line segment is steeper, line NK or line PM?”

• Unit 9, Lesson 9.3, About Congruence, Problem 2, students develop understanding that two- dimensional figures are congruent if the second can be obtained through a sequence of transformations (8.G.2). “Small grid squares are one square unit of area. Use patty paper if needed. 2) 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?”

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. For example:

• Unit 3, Lesson 3.3, Practice 8, Problems 20-21, students demonstrate understanding of square and cube roots to represent solutions in the form x^2=p and ^3x=p (8.EE.2). “Explain each relationship with a sketch and numbers. Use squares, square roots, cubes, and cube roots. 20) The area of a square and its side lengths; 21) The volume of a cube and its edge lengths.”

• Unit 4, Lesson 4.1, Analyzing The Pool Problem, Problems 1-9, students demonstrate understanding of comparing properties of two functions represented in different ways (8.F.2). “3) When does the Border Pattern have more squares than the Water Pattern? 5) Write the number of squares for each pattern for Poll 20. 7) Explain what (0, 4) and (0, 0) represent in the context of The Pool Problem. Where are these points found on the graphs? 9) Does the Water Pattern grow at a constant rate? Explain. Does it represent a proportional relationship? Explain.”

• Unit 7, Lesson 7.1, Practice 1, Problem 6, students demonstrate understanding that solutions to a system of two linear equations correspond to points of intersection of their graphs (8.EE.8a). “Sketch the graph of a system of linear equations with exactly one solution such that both lines have negative slopes.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 8 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

According to the MathLinks Program Information, in Focus, Coherence and Rigor, “In MathLinks, we thoughtfully develop new procedural skills and provide opportunities for students to gain fluency throughout the year. Skills practice in each unit is found in the Student Packets in the following ways: Practice pages – These pages support concept development. Review activities – These pages often include skills practice. Spiral Review – These pages have distributed practice of prior skills. Alge-Grids – These puzzles, created by Carole Greenes and Tanner Wolfram, give students practice with expressions and exponents. They appear in half of the Spiral Reviews. READY-X – These puzzles, created by Carole Greenes and Tanner Wolfram, give students practice with single-and multi-variable equations. They appear in half the Spiral Reviews.”

In addition to what is in the student packets, teachers have access to additional support for developing procedural skill and fluency. “Grade-level skills practice is in each unit, as well as practice to fill in gaps. Both can be found on the Teacher Portal in Other Resources in the following ways: Essential Skills – This entire section reviews skills and concepts important for success in a given unit. Activity Routines such as Big Square Puzzles, Open Middle Problems, and Four-in-a-Row games are also in these sections for some units. They provide a practice alternative to “drill and kill.” Extra Problems – Skills practice by lesson is available for all units. Non routine Problems – In addition to skills practice that is embedded in nonroutine problems, Big Square Puzzles, Open Middle Problems, and Four-in-a-Row games are located in this section for some units.” A table is provided that identifies multiple “examples of fluency work” throughout the lessons.

Materials develop procedural skills and fluency throughout the grade level. (Note: Lesson notes come after the workbook page in the Teacher Edition.) For example:

• Unit 3, Lesson 3.1, Lesson Notes S3.1: Investigating Two Exponent Patterns, Problems 1-17, students apply the properties of integer exponents to create equivalent numerical expressions (8.EE.1). “Copy each expression below and fill in the table. 1) 8^2\cdot8^3” A column is provided for students to write Factors (as a multiplication expression) and Exponent form (b^n). Teachers have prompts such as, “How is this expression the same as the one before it?” 

• Unit 5, Lesson 5.2, Getting Started, Problems 3-5, students practice determining the rate of change and initial value of the function using multiple representations (8.F.4). “For each function rule below, complete the table, graph the line, and identify the slope and y-intercept.” Function rules are provided with x values of 0, 1, 2, -1 for each rule. 

• Unit 8, Lesson 8.3, Lesson Notes S8.3: Watering Cans, students follow teacher directions to practice creating equations and interpreting the solutions based on the situation (8.EE.7b). Depending on what they spin, there are 16 possibilities for finding when two equations are equal. For example, Can B with Can F yields “1500-75x=1200-100x; x=-12” Teacher prompt, “How would you explain the solution to Dion?” Since it’s a negative answer, there’s no real solution. “Can F starts with less water and drains faster than Can B.”

Materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. For example:

• Unit 1, Teacher Portal, Other Resources, Extra Problem Reproducibles, Lesson 8-1.2, Problem 2, students find the volume of spheres (8.G.9). “Eduardo has a fruit bowl on his kitchen counter. Find the volume of each piece of fruit in the bowl. (Measurements are approximations.) Use \pi=3.14. a) An orange with a diameter of 8 cm. b) A plum with a radius of 2 cm. c) A guava with a diameter of 7 cm.” 

• Unit 7, Lesson 7.1, Problems 4-5, students graph systems of two linear equations and identify the number of solutions (8.EE.8b). “For the system of equations below, first make sure the equations are in slope-intercept form. Then graph the lines, determine the number of solutions, and write the solution(s), if any. 4) 3y=9x+3 // y=3x-5;  5) y-5=x // 2x+y=-4.” Students are given a coordinate grid to graph the solution. 

• Unit 10, Spiral Review, Problems 3a-i, students solve linear equations in one variable (8.EE.7). Examples include: c) -x-5=4(x+2)-2; e) -x-4-x=-3(-2x-2); h) \frac{1}{4}x-6=\frac{1}{2}x+2.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

According to the MathLinks Program Information, in Focus, Coherence and Rigor, “Problem solving is an important driver of instruction within MathLinks courses. In MathLinks, we include engaging mathematical problems and applications with accessible entry points for all students, multiple approaches or solutions, and extensions to challenge and enrich. All units begin with an Opening Problem, which introduces a concept or establishes a ‘need to know.’ In many cases, students require more instruction throughout the unit before they are fully prepared to bring the problem to its conclusion. Substantial problems exist throughout the units as well.” A table is provided that identifies multiple “examples of problem-solving lessons” throughout the lessons and additional resources such as tasks and projects. 

Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Though, in many cases, problems labeled “non-routine” in the materials are actually routine problems since there is only one solution path to an expected answer, even though the context of the problem may be novel. For example:

• Unit 1, Teacher Portal, Other Resources, Non-Routine Problems Reproducibles, Solid Puzzlers, Problems 1-3, students apply their knowledge of volume in a non-routine problem (8.G.9). “1) A sphere and a cone have the same volume. Each figure has a diameter of twelve inches. What is the height of the cone? 2) A cone and cylinder have the same radius and height. What is the value of the ratio of the volume of the cone to the volume of the cylinder? 3) A cone and cylinder have the same radius and different heights. How must the heights of the cone and cylinder compare so that their volumes are equal?”

• Unit 2, Lesson 2.2, The Club and the Box, Problem 1, students apply their understanding of Pythagorean Theorem in a routine real-world problem (8.G.7). “1) Dorie is an avid golfer. Lorie has recently taken up the sport and Dorie wants to send Lorie one of her old golf clubs. Dorie’s club length is 45 inches, and she needs to figure out the smallest box she can buy to mail it to Lorie so that postage is not too high. Dorie finds a box (pictured below), but thinks it’s too small. She tells Lorie the dimensions, and after making some calculations, Lorie thinks the club will fit. Draw on the figure, show calculations, and write a sentence to support either Dorie’s claim or Lorie’s claim.” The box has dimensions of 40 in \times 20 in \times 10 in. 

• Unit 8, Lesson 8.3, Algebra Applications, Training for a Marathon, students use their knowledge of linear functions to compare initial values, pace, and determine when they are equivalent in a routine real-world problem (8.F.4). “Marathon runners keep track of their progress by measuring “pace” (minutes per mile). Robin and Jacob are training for an upcoming marathon. They don’t usually train together because their paces are so different, but decide to train together today. Jacob says, “I’ll give you a one-hour head start. Let’s see when I catch up to you. Robin is an average runner. Below is a table of his training at a pace.” Table of values is given. “Jacob is an excellent wheelchair athlete. To the right is a graph of his training at a constant pace. How is Jacob’s head start reflected on the graph? Write each of their initial values. Write each of their paces in minutes per mile. Write equations to represent each of their training paces. Use substitution to solve this system. Then state the time it takes for Jacob to catch up to Robin and at what mile that occurs.” 

Materials provide opportunities for students to independently demonstrate multiple routine and non- routine applications of the mathematics throughout the grade level. For example:

• Unit 1, Teacher Portal, Other Resources, Nonroutine Problems Reproducibles, Interior Angles of Polygons, students apply facts about angles to critique an argument in a non-routine problem (8.G.5). “Kendall thinks that the sum of the interior angle measures of a triangle is 180\degree. He claims that the sum of the interior angle measures of a hexagon must equal 360\degree because a hexagon has twice as many vertices as a triangle. Therefore, a hexagon must have twice as many degrees as a triangle. Fill in the table. Sketch some of the polygons and explore how drawing triangles in the polygon interiors can lead to conclusions about the sum of the interior angles for each polygon. Pay special attention to creating triangles by drawing all diagonals from one vertex of a polygon (2nd row in the table).” Students are given a table labeled with Number of Sides, Number of Diagonals From Any One Vertex, Number of “Triangles” Formed, and Sum of Interior Angles. “Is Kendall correct? Critique Kendall’s reasoning.” 

• Unit 5, Lesson 5.3, Applications and Extensions, Rectangle Paradox: A Fresh Look, students are challenged to prove two rectangles are not the “same” in a non-routine application of slope (8.F.B). “Do you remember this problem from Unit 2? Rectangle I is cut apart and rearranged to form Rectangle II. If you find the area of these rectangles, you’ll notice they are different. How can that possibly happen? Verify the rectangle areas. Rectangle I: length = ___u; width = ___u;  area = ___ u^2; Rectangle II: length = ___u;  width = ___ u;  area = ___u^2. As you can see, the areas are very close! If you solved the problem in Unit 2, you proved that Rectangle I actually cannot be rearranged to form Rectangle II. And, in doing so, you probably used the Pythagorean theorem, which is one of the big ideas of that unit. Your job is to use something you’ve learned in this unit to prove in a different way that Rectangle I cannot be rearranged to form Rectangle II. In other words, what’s wrong with Rectangle II?”

• Unit 7, Lesson 7.1, Practice 3, students apply their knowledge of systems to graph two functions to compare data, then solve with substitution to determine equivalence in a routine real-world problem (8.EE.8c). “Naomi and Karolina are saving for a skateboard. Naomi has $100 in the bank and will save $30 each month. Karolina has $40 in the bank and will save $45 each month. 1) Complete the table below, graph the data and write the input-output equations.” A table shows a columns under both Naomi and Karolina with Month #(x) and Total saved in $(y). Beneath these are rows with numbers 0-7 and a final row with y=. There is also a graph of the first quadrant. “2) Who is saving at a faster rate? Justify your answer by referring to some problem 1 representations. 3) During which month(s)... a) does Naomi save more money?; b) does Karolina save more money?; c) do they have the same amount of money? What do you notice about the table entries this month? What do you notice about the graphs this month? 4) Use substitution to write one equation in x equating Naomi’s and Karolina’s savings. Use this equation to verify the month at which they have the same amount of money. State your answer in a short sentence.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 8 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.

All three aspects of rigor are present independently throughout each grade level. For example:

• Unit 2, Lesson 2.2, A Right Triangle Investigation, students develop conceptual understanding of Pythagorean theorem (8.G.B). Students draw two different right triangles on a grid, then complete a chart for both answering; “Length of the shorter leg; Length of the longer leg; Area of the square on the shorter leg; Area of the square on the longer leg; Area of the square on the hypotenuse; Length of the hypotenuse.” This is followed with “Write a conjecture about the relationship between the area of the square on the hypotenuse and the area of the square on the legs of a right triangle.” 

• Unit 5, Teacher Portal, Nonroutine Problems Reproducibles, Mixed Problems, Problem 6, students develop fluency in being able to define parts of a function (8.F.A). “Consider the equation y=\frac{3}{4}x+\frac{1}{4}. Select ALL true statements below. a) y is a function of x; b) The graph of the equation is a line; c) The y-intercept is \frac{3}{4};  d) The graph of the equation is decreasing; e) The slope is \frac{3}{4};  f) When x=1, y=1: g) When x=3, y=3; h) When x=5, y=4.

• Unit 8, Lesson 8.3, Practice 6, Problem 1, students apply understanding of writing and solving linear equations (8.EE.7). “Ada has $84.75 and is saving $58.50 per week. Thabo has $177.25 and is saving $40 per week. After how many weeks will they have the same amount of money?”

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout each grade level. For example:

• Unit 3, Lesson 3.2, Practice 5, Problems 9 and 11, students apply conceptual understanding of large number representation and develop procedural skill in scientific notation in real-world problems (8.EE.4). “Dr. Jerry Buss purchased the Los Angeles Lakers basketball team in 1979 for approximately $67.5 million. At his death in 2013, the team was reportedly worth about $1 billion. a) Write each dollar amount as a single digit times a power of 10. b) Write each value using scientific notation. c) By approximately what factor did the team value grow over that time period? 11) Frazier was trying to multiply 5,400,000,000,000,000\times75,000 using his phone, but when he input the first number he got a message that 15 digits is the maximum allowed. Describe how he can multiply these numbers, and write the result using scientific notation.” 

• Unit 6, Lesson 6.2, Obesity Rates by State, Problems 1-5, students apply conceptual understanding of linear functions to determine a line of best fit for interpreting data (8.SP.A and 8.F.4). “1) Estimate (draw) a line of best fit on the graph below and write its equation. Since the y-intercept is above the graph, you may want to use a ruler. 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? 4) Use your  equation to predict the obesity rate in a state where 50% of the population exercises. 5) The CDC recommends to governors of all states to set a goal of 20% for their obesity rates. What does your model predict that the exercise rate should be to meet this goal?”

• Unit 7, Teacher Portal, Other Resources, Projects Reproducibles, Staircase Slopes, students develop procedural skill to apply understanding of slope in a real-world problem (8.F.B). “Stairs are made up of treads, which are the horizontal parts you step on, and risers, which are the vertical boards holding up the tread. Laws regulate the height of risers and require risers to be of uniform height. 1) The slogan formula for finding slope is \frac{rise}{run}. On a staircase, what measurement corresponds to the “rise,” and what measurement corresponds to the ‘run’? 2) Find at least three different sets of staircases. For each staircase, take appropriate measurements of the riser and the tread for several stair steps. Describe the measurement tools and units you used. Complete steps 3 and 4 for each of the staircases you chose. 3) Create a table to record your tread and riser measurements for each step as well as the \frac{rise}{run} ratio as a fraction. 4) Compare the ratios for different steps. Are these ratios equivalent? If not, find a ratio that appropriately represents the entire staircase. Explain how you figured out this value. 5) Write a short paragraph to (1) describe what this value determined in problem 4 represents in the context of each staircase, and (2) compare these values for your different staircases and use descriptive words similar to those used in class. Challenge: Research laws that regulate the height of risers in your community. Are the staircases you measured ‘built to code’?”

• Unit 10, Essential Skills Reproducibles, Practice with Rigid Motions, Problems 1-4, students demonstrate conceptual understanding and develop fluency with transformations (8.G.A). “Demonstrate how to obtain the image (unshaded figure) from the pre-image (shaded) by a sequence of translations, rotations, and reflections in TWO different ways. Use patty paper as needed.” Students have 3 problems to describe. “4) Explain how translations, rotations,and reflections are related to congruence.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 8 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. 

Mathematical Practice Standards are identified in the Teacher Edition and Student Packets. The Teacher Edition has Teaching Tips that include abbreviated descriptions of the Standards for Mathematical Practice and specific examples and explanations that describe how the practices are developed within the unit. 

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the units. Examples include:

• Unit 1, Review, A Big Puzzle, Problems 1-31, students make sense of the problem by looking for entry points to find the measure of angles. (MP1) “Use the figure below for all problems. For problems 1-8, name an angle that is: 1) adjacent to \angle m; 2) vertical to \angle r...” For 9-31, students are given a figure created by parallel, perpendicular, and transversal lines. “Find the angle measures in any order. Notice that \left\vert\angle a\rvert, \lvert\angle h\rvert, and \lvert\angle z\rvert are given.” Students find angles a-z. 

• Unit 5, Review, Open Middle Problems: Linear Functions, Problems 1-3, students analyze and make sense of problems and plan a solution pathway to complete the puzzle (MP1). “1) Using the digits 0 – 9, no more than once each, create a table of values such that the points graphed are all along the same line. For your line, write the: slope, y-intercept, equation. There is a line that goes through the origin and point (a, b). Find values for a and b to satisfy each condition for problems 2 – 3. Use only the digits 1 – 9, no more than once each for each problem. 2) The greatest positive slope possible. The equation for this line is; 3) The least positive slope possible.The equation for this line is.”

• Unit 8, Teacher Portal, Tasks Reproducibles, Soccer Club Orders, students analyze and make sense of real-world problems’ information and questions asked (MP1). “The table below shows the numbers of jerseys and sweatshirts ordered for two boys’ soccer teams. All jerseys cost the same amount and all sweatshirts cost the same amount. Jerseys do not cost the same as sweatshirts.” A table is given showing orders for Team A and Team B. “Coach Taniel represents the orders above with the following system of equations. Team A: 3x+3y=228; Team B: x+4y=202). 1) Describe in words what x and y represent in this situation. 2) What is the cost of 1 jersey plus 1 sweatshirt? Justify using words and numbers. 3) If the system of equations was graphed on a coordinate plane, what would be the point of intersection of the two lines? 4) What do the coordinates of the point of intersection represent in this situation? 5) Find the total cost for Team C, if the prices remain the same. Show all work.” 

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the units. Examples include:

• Unit 3, Review, Why Doesn’t It Belong?: The Algebra of Exponents and Roots, students reason abstractly and quantitatively by attending to the meaning of the quantities and using properties flexibly with exponents and roots (MP2). “For each set below, choose at least two of the four entries and explain why each doesn’t belong with the others.

  • Set 1: Simplify each expression. A)  (x^2)^2  ; B) x^8\cdot x^{-4} ; C) \frac{x^6}{x^8}; D) \frac{x^{-4}}{x^{-8}}

  • Set 2: Write each expression with a base to a non-negative exponent. A) x^4\cdot x^{-4}; B) \frac{1}{x^{-4}}; C) (x^2)^{-1}; D) x^{-4}

  • Set 3: Simplify the two expressions and solve the two equations. A) \sqrt{9}; B)  \sqrt[3]{-27}; C) x^2=9; D) x^3=27”

• Unit 5, Lesson 5.3, Practice 8, Problem 10, students reason abstractly to show relationships in problem situations related to linear functions (MP2). “A line goes through the origin and the points (0, 9) and (6, -21). Without graphing, write the equation of this line in slope-intercept form.”

• Unit 7, Lesson 7.1, Using Substitution, Problems 4-9, students use quantitative reasoning to decontextualize a situation and represent it symbolically to manipulate linear functions (MP2). “Kim and her friend Jordan meet for lunch. Kim tells Jordan about the 100 Mile Walking Challenge she’s been doing for a while. ‘I already have 40 miles, and starting tomorrow, I’m going to walk 8 miles per day,’ says Kim. ‘You should join the challenge.’ Jordan accepts and says, ‘Okay, you’re way ahead of me, so I’m going to walk more miles per day to try to catch up.’ Students complete a table for days of walking from 0-10. “5) From the table, on what day after their lunch meeting does each succeed in the challenge? 6) Record what the variables mean. Write equations that model walking over these days for each friend. Substitute 100 miles into each equation. Solve for the number of days. 7) Why do the solutions to problem 6 support your answer to problem 5? 8) If Kim and Jordan each continue to walk at their same pace, on what day have they walked the same number of miles? How many miles is this? 9) Use substitution and the two equations from problem 6 to write one equation in x. Then check to see if the x-value (day number from problem 8) is a solution to this equation.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 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.

Mathematical Practice Standards are identified in the Teacher Edition and Student Packets. The Teacher Edition has Teaching Tips that include abbreviated descriptions of the Standards for Mathematical Practice and specific examples and explanations that describe how the practices are developed within the unit. 

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Students construct viable arguments and critique the reasoning of others as they work with the support of the teacher and independently throughout the units. Examples include:

• Unit 3, Lesson 3.1, A Third Pattern: the Quotient Rule for Exponents, Problem 19, students critique the reasoning of others about exponent rules (MP3). “Patti thinks that a base number to a negative power must result in a negative value. Is Patti correct? Explain.” 

• Unit 5, Spiral Review, Problem 6, students construct viable arguments to explain irrational numbers (MP3). “Olivia and Addie agree that the value of \sqrt{30} is between 5 and 6. Olivia thinks that \sqrt{30} is closer to 5. Addie thinks that \sqrt{30} is closer to 6. a) Which student do you think is more accurate? b) What would you say to the other student to help her understand her error?”

• Unit 6, Lesson 6.2, Analyzing Education Data, Problem 4, students construct viable arguments as they create their own conjectures (MP3). “Is it reasonable to use the y-intercept to make predictions about the income of a person with zero years of education? Explain.”

• Unit 10, Lesson 10.2, Practice 3, Problems 3-6, student critique the reasoning of others by performing error analysis for angle relationships and similarity (MP3). “For problems 3 – 6, critique each student’s statement. 3) Triangle 1 has a 45\degree angle and a 50\degree angle. Triangle 2 also has a 45\degree and a 50\degree angle. Jacob thinks that these triangles must have congruent third angles. 4) Maya dilates a pre-image figure to obtain a similar image figure. Maya thinks that corresponding parallel sides from the pre-image figure remain parallel in the similar figure. 5) Drew thinks that corresponding angles 1 and 2 must be congruent (given two non-parallel lines with a transversal). 6) Ayla thinks that a dilation of a pre-image alone may result in a similar image figure.”

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

Mathematical Practice Standards are identified in the Teacher Edition and Student Packets. The Teacher Edition has Teaching Tips that include abbreviated descriptions of the Standards for Mathematical Practice and specific examples and explanations that describe how the practices are developed within the unit. From the Applying Standards for Mathematical Practice (SMP) section in the Teacher Edition, “[All Lessons] Students record mathematics vocabulary as it is introduced in lessons. They use precise language in writing and exercises. Precise definitions are located in the Student Resources section in the back of the unit.” This is true for all units.

There is intentional development of MP6 to meet its full intent in connection to grade-level content and the instructional materials attend to the specialized language of mathematics. Students attend to precision as they work with the support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 1.1, Practice 1, Problems 1-5, students calculate accurately and efficiently as they express numerical answers with a degree of precision appropriate for the problem context. “Find the volume of each cylinder described below. 2) The base is pictured below. The diameter is given in millimeters. The height is 20 mm. Use \pi=3.14. 5) The base radius is 3\frac{1}{2} cm. The height is 4 cm. Use \pi=\frac{22}{7}.”

• Unit 3, Review, Match and Compare Sort: Solving Equations, Problems 1-3, students differentiate between pairs of words that might be closely related, and write at least one detailed set of reasons of similarities and differences. Students “connect concepts to vocabulary words and phrases, 1) Individually, match words with descriptions. Record results into a table. 2) Partners, choose a pair of numbered matched cards and record the attributes that are the same and those that are different.” A Venn diagram is provided for students to place vocabulary words in boxes and explanations in circles that list similarities and differences. “3) Partners, choose another pair of numbered matched cards and discuss the attributes that are the same and those that are different.” The vocabulary for this unit is cube of a number, cube root of a number, power rule, product rule, scientific notation, exponent notation, base exponent. Match and Compare Sorts are included in many units.

• Unit 4, Review, Vocabulary Review crossword, each puzzle clue focuses on specialized language, “2) where lines cross is a point of ___, 8) A measure of the steepness of a line, 11) lines that never meet.” Each unit review includes a vocabulary crossword.

• Unit 7, Lesson 7.1, Practice 1, Problems 4-5, students attend to precision to solve systems of equations and determine the number of solutions. “For the systems of equations below, first make sure equations are in slope-intercept form. Then graph the lines, determine the number of solutions, and write the solution(s), if any. 4) 3y=9x+3; y=3x-5 5) y-5=x; 2x+y=-4.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 8 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.

Mathematical Practice Standards are identified in the Teacher Edition and Student Packets. The Teacher Edition has Teaching Tips that include abbreviated descriptions of the Standards for Mathematical Practice and specific examples and explanations that describe how the practices are developed within the unit.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Students look for and make use of structure as they work with the support of the teacher and independently throughout the units. Examples include: 

• Unit 3, Lesson 3.1, Practice 1, Problem 15, students explain the structure of exponent rules to demonstrate understanding of a concept. “Josue looked at problem 11 and said, ‘Even if I mixed up these two exponent rules, I could still get it right.’ Explain what Josue meant.”

• Unit 4, Lesson 4.1, Getting Started, Problem 1, students look for patterns to complete a table of values, find rate of change, and describe an input-output rule. “Fill in the missing numbers and blanks based on the suggested numerical patterns. In the tables below, the x-value is considered the input value (independent variable) and the y-value is the output value (dependent variable). a) Rate of change: for every increase of x by 1, y increases by ___; b) Input-output rule (words): multiply the x-value by ___ to get the corresponding y-value; c) Input-output rule (equation): y = ___. When x = 0, y = ___.”

• Unit 7, Lesson 7.2, Solving Equations with Balance 1, students see complicated objects as being composed of smaller, simpler objects. “Follow your teacher’s directions for (1) – (4).  When solving equations using cups and counters: Build each equation; Think: Can I do anything to either side (individually)?; Think: Can I do anything to both sides (together)?; Continue the process until the equation is solved; Write the solution and check it using substitution; Make a drawing of the process; Copy each equation; Solve by building; Record drawings; Check solutions; Drawings may vary.” 

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Students look for and express regularity in repeated reasoning as they work with the support of the teacher and independently throughout the units. Examples include: 

• Unit 2, Lesson 2.3, A Rational Numbers Investigation, Problems 1- 4, students notice repeated calculations to generalize why decimals repeat or terminate. “Continue each pattern below, and describe it in words. Use a calculator to check as needed. 1) Ninths as decimals; Pattern description; 2) Elevenths as decimals; Pattern description; 3) Sevenths as decimals; Pattern description; 4)The fraction \frac{s}{b} poses the division problem a\divb. Reason why any fraction that is converted to a decimal by division must have a pattern for which a repeat bar can be used. (Recall that \frac{1}{2}, though terminating, can be written as (0.5=0.50000…=0.5\bar{0}).”

• Unit 6, Lesson 6.1, Practice 1, Problem 1, students use repeated reasoning to discover a strategy that will work to identify data associations without context. “Look at the sets of (x, y) ordered pairs below, all without contexts. Predict the kind of association each has, if any, by observing patterns in the data. Graph points to verify predictions. Set 1: (0, 5) (0.5, 4) (1, 4.2) (1.5, 4) (2, 3.5) (3, 3.8) (3.5, 2.6) (4, 1.9) (5, 1.8).” Students investigate three sets of data.

• Unit 10, Lesson 10.1, Getting Started, Problems 2-6, students use repeated calculations to generalize rules for square roots. “Simplify each expression below. 2a) \frac{\sqrt{36}}{\sqrt{9}}, 2b)\sqrt{\frac{36}{9}}, 3a) \frac{\sqrt{81}}{\sqrt{9}}, 3b) \sqrt{\frac{81}{9}}, 4a) \frac{\sqrt{64}}{\sqrt{4}}, 4b) \sqrt{\frac{64}{4}}, 5a) \frac{\sqrt{100}}{\sqrt{4}}, 5b) \sqrt{\frac{100}{4}}, 6) Based on problems 2-5 above, make a conjecture by filling in the blanks below. Then explain the conjecture in words. \frac{\sqrt{m}}{\sqrt{n}}=\sqrt{\frac{m}{n}}.”