8th Grade - Gateway 2

The instructional materials for Core Curriculum by MidSchoolMath 5-8, Grade 8 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Examples of problems and questions that develop conceptual understanding across the grade level include:

• In 8.EE.B.5, Space Race, students work together to answer, “Which ship will get to Candoran first?” “The data provided are the distometer display, which shows a distance-time equation, and the radar display which shows a distance-time graph.” 

• In 8.F.A.1, Flight Functions, the Immersion Situation states, “In Flight Functions, Darla Macguire is training to become an air traffic controller. During training she states that a plane can be in two locations at once. Her trainer, Anita, stops her to explain this is impossible and asks her if she knows what a mathematical function is. Macguire states that she does know about functions, and they have a quick discussion. Anita gives her the task to determine which displays are functions before continuing her training. The data provided is an image of the ATC Radar Function Assessment.” In Data & Computation, the teacher asks students several questions, “What is a function? Is there only one output for each input? If you draw a vertical line, is there more than one y-variable for any x-value? What does it mean if there are two output values for an input?” During the Teacher Instruction several examples are provided to identify examples and nonexamples of functions.

• In 8.EE.B.6, Ghost Island, the Practice Printable includes, “Aboard the ship, Isosceles, Captain Mary Read and Sailing Master Bonny Anne are headed to find more treasure. They have departed Isla Fantasma and have set a course. They hope to arrive at Isla Pinos. It is thought to hold the hidden treasure of One-Eyed Whitebeard. Use the readings on the map to help Bonny Anne determine whether the Isosceles is on course to hit Isla Pinos.” Students are given two maps, each showing a triangle - one map is between Fantasma and the Isosceles, the other between Fantasma and Pinos. 

• In 8.EE.C.8a, Show Me the Money, the Teacher Instruction includes questions to further students thinking such as, “What information are you given? Can you graph an equation of each offer? Where would you start the graph for the Madison contract? What would be the total earnings for 16 games with Madison? Where would you start the graph for the Tallahassee contract? What would be the total earnings for 16 games with Tallahassee? Do the lines intersect? Where? What can we conclude about which offer would be a better deal for Cage based on his past play record?"

Examples where students independently demonstrate conceptual understanding throughout the grade include:

• In 8.F.A.2, Happy Trails, Practice Printable, Question 2 states, “a) Draw Function D with a rate of change (slope) of and a y-intercept of 1.  b) For Function E, write a linear equation with a greater rate of change (slope) than Function D and a y-intercept that is below the x-axis.  c) Create a table of values that shows Function F is proportional and has a rate of change (slope) of -3.”

• In 8.F.A.3, Le Monsieur Chef, Practice Printable, Question 1d, students answer true/false questions about functions shown on a graph such as “As $$x$$ increases by 1, $$y$$ always increases by 2.”

• In 8.G.A.4, Dakota Jones and the Hall of Records, Practice Printable, Question 2 states, “A transformation is made to $$\triangle$$ABC to form $$\triangle$$DEF (not shown). Then another transformation is made to $$\triangle$$DEF to form $$\triangle$$GHJ. Describe these transformations. Then tell whether ABC and GHJ are congruent, similar, or neither. Explain why.”

• In 8.NS.A.1, Warp Speed, Practice Printable, students show that irrational numbers do not have a fractional equivalent. Students consider 10 terms in a table to determine if they are rational or irrational, then justify their answer by creating a fractional equivalent (or none). Terms include, “$$-72; \frac{25}{\sqrt{4}}; 0.414141…$$; $$\sqrt{121}$$; $$\sqrt{14}$$; -0.3333…; $$\sqrt{50}$$; 1.25; $$\frac{\pi}{2}$$; 0.68.”

The instructional materials for Core Curriculum by MidSchoolMath 5-8, Grade 8 meet expectations for attending to those standards that set an expectation of procedural skill and fluency. 

The instructional materials develop procedural skill and fluency throughout the grade level in the Math Simulator, examples in Teacher Instruction, Cluster Intensives, domain-specific Test Trainer Pro, and the Clicker Quiz. Examples include:

• In 8.EE.C.7b, Petunia’s Pickle Problem, there are five examples for the teacher to use during instruction, “Let's see another example. Solve: -$$-3(-4p-6)+12=-15+9(\frac{1}{3}p-1)$$. As always when solving an equation, our goal is to isolate the variable, which in this case is p. To do that, we start by simplifying both the left and the right sides of the equation. We first need to use the distributive property to get our equation in a more usable form. On the left we must remain  aware of our negative values and give our terms the appropriate signs. $$12p+18+12=-15+3p-9$$. Now we can combine like terms. $$12p+30=-24+3p$$. Finally we can use inverse operations to get our variable terms on one side and the constants on the other. $$19p=-54$$. Now we divide both sides by 19 to find the value of p. $$p=-6$$.”

• In 8.F.B.5, Twin Tactics, Teacher Instruction, “Let’s try another example. As we read each statement, let’s sketch that part of the graph. At the start of a softball game, there were 25 people sitting on bleachers. [Plot the point (0, 25).] Inning 1, there were 45 people on the bleachers. [Plot the point (1, 45).] The number of people on the bleachers remained constant for three more innings. [Plot the point (4, 45).] 0 people left the bleachers in the 5th inning. [Plot the point (5, 35).] A family of 5 sat down during the 6th inning. [Plot the point (6, 40).] There was no movement until the 8th inning when 20 people left. [Plot the point (8, 20).] Inning 9, there was no one sitting in the bleachers. [Plot the point (9, 0).]” WIthin one problem, students have multiple opportunities to sketch segments on a graph that demonstrate the features described. 

• In 8.G.B.7, The Road Less Traveled, Teacher Instruction includes engaging situations to practice the procedural steps of Pythagorean Theorem, including the shortest route, a rectangular prism, and a soccer field, “We know this is a rectangular field, so that makes the corners right angles. That means we can use the Pythagorean Theorem. The longest side of a right triangle is always opposite the right angle. We see here that 120 is opposite the right angle, so that will be side c. So we can substitute 96 for a (or b) and 120 for c, and simplify.”

Examples of students independently demonstrating procedural skills and fluencies include:

• In 8.EE.C.7b, Petunia’s Pickle Problem, Practice Printable, students solve multi-step linear equations with rational coefficients in a variety of contexts including finding the length of line segments, the measure of vertical angles, break-even points, and practice without context such as Question 4, “$$-4x+15-x=2(6-2x)-x$$”.

• In 8.F.B.5, Twin Tactics, the Practice Printable provides 14 questions related to analyzing the functional relationship between two quantities on a graph and 11 more questions to sketch a graph that’s described, such as Question 3, ‘Ben washed dishes when he got home from school. Using the details provided, sketch a graph depicting Ben’s experience. Ben took two minutes to fill the sink to half-full. He then washed the dishes for 8 minutes. He took about 3 minutes to drain half the water out and to fill it back up to half-full with new warm water. It took him four minutes to finish washing the rest of the dishes. Then he drained the water. It took Ben 20 minutes in total from start to finish.” 

• In 8.G.B.7, The Road Less Traveled, the Practice Printable provides 12 opportunities for students to use Pythagorean Theorem in a variety of situations including three-dimensional prisms, perimeter, travel routes, and straightforward practice such as Question 1, “For each triangle, find the length of the missing side.”

The instructional materials do not develop procedural skill or provide enough opportunity for students to demonstrate procedural skill with the following standard.

• In 8.EE.C.8b, Mars Rocks!, Teacher Instruction includes two examples for number of solutions, one example of solving a system by substitution, and one example of solving a system by elimination. The Practice Printable provides six problems for students to independently identify the number of solutions and four problems to solve algebraically. Due to limited instruction and practice, students do not develop procedural skill with solving systems of two linear equations in two variables algebraically and estimating solutions by graphing the equations.

The instructional materials reviewed for Core Curriculum by MidSchoolMath 5-8, Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications include single and multi-step problems, routine and non-routine problems, presented in a context in which mathematics is applied. 

Examples of students engaging in routine application of skills and knowledge include:

• In 8.EE.C.8c, Training Day, Practice Printable, Question 3 states, “The Sno-Cone Casa sells snow cones during the summer months. They charge $4 per snow cone, or there is an option to join the Sno-Cone Club, which charges a one-time fee of $20 and $1.50 for each snow cone. For what number of snow cones (c) is the price (p) the same, whether or not you join the club? What is the price?” 

• In 8.G.B.7, Road Less Traveled, Practice Printable, Question 4 states, “How long are the poles for the teepee, assuming an extra 3 feet for the top?” Students use an image of a teepee given the diameter of the base and the height. The slant height is a variable, and students use the Pythagorean Theorem to solve.

• In 8.F.B.5, Twin Tactics, Practice Printable, Question 3 states, “Ben washed dishes when he got home from school. Using the details provided, sketch a graph depicting Ben’s experience: Ben took two minutes to fill the sink to half full. He then washed the dishes for 8 minutes. He took about 3 minutes to drain half the water out and to fill it back up to half-full with new warm water. it took him four minutes to finish washing the rest of the dishes. Then he drained the water. It took Ben 20 minutes in total from start to finish.”

• In 8.SP.A.1, Cholera Outbreak!, Practice Printable, Question 2 states, “A survey group conducted a study to determine if there is an association between the age of a person and the average number of emojis used per text. The survey results are below.  a) Create a scatterplot on the graph, using the data in the table. b) What type of association do the two variables seem to have? c) Does there appear to be an outlier? d) Based on the data, what can you say about age and the number of emojis used per text?”

Examples of students engaging in non-routine application of skills and knowledge include:

• In 8.EE.A.3, Malaria Medicine, Practice Printable, the Introduction Problem states, “How many times as great should the dosage be? Before taking her human malaria studies “underground,” Doctor Sofia Martín was also attempting to cure malaria in horses. As you know, she has been conducting research using mice infected with the dreadful disease. She has determined the correct spiroindolone dose for the mice and has cured many of them. Doctor Martín believes she can use what she knows about the mice to help find the correct dosage for horses. She believes the dosage is proportional to the number of infected cells. Below, find one of her lab reports. Help her determine how many times as great the horse dose will be compared to the dose for mice.” The infected cell count is given in scientific notation.

• In 8.EE.C.8b, Mars Rocks!, the last question on the Practice Printable states, "Braydon ran at a rate of 6 mph on his treadmill and then took a walk outside at a rate of 2 mph. He went a total distance of 10 miles and it took him 2 hours. How long did he run on the treadmill and how long did he walk outside?”

• In 8.G.A.2, Knee Replacement, the Practice Printable presents, “Dr. Perkins and his assistant are preparing a knee implant for a patient. They will use the voice-activated Transforma software to map the MRI image onto the Replacement Image. Study the images on the screen and determine the commands (mathematical transformations) that Dr. Perkins needs to tell Transforma to map the MRI image onto the replacement image.”

• In 8.G.C.9, The Dawn of Anesthesia, Practice Printable, Question 5 states, “In the figure below, the cylinder is solid and water is to fill the rest of the figure. What is the volume of water that will fill the figure? Round your answer to the nearest tenth.” Students must calculate the volume of the excess space inside a cube that contains a solid cylinder.

he instructional materials reviewed for Core Curriculum by MidSchoolMath 5-8, Grade 8 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

Examples of the three aspects of rigor being present independently throughout the materials include:

• In 8.SP.A.3, The Slope of Sprouts, students develop conceptual understanding of linear models to solve problems in the context of bivariate measurement data. In the Practice Printable, Question 2 states, “For each problem below, an equation is given that models the linear relationship of two variables. Interpret the slope and y-intercept (if reasonable) within the context of the two variables.” Students are given data: “$$y = 19.35x - 870$$; $$x$$: Temperature ($$\degree F$$), $$y$$: Ice Cream Sales ($)” to “describe the meaning” of the slope and the y-intercept.”

• In 8.G.A.3, Tile Transformation, students use procedural skills to write the coordinate rule for each transformation: dilate, rotate, translate and reflect. In the Practice Printable, Questions 1-9, all involve identifying “what happens to (x, y) for each transformation.” Examples include: “9) Rotate quadrilateral WXYZ $$90\degree$$counterclockwise about the origin. Label the image W'X'Y'Z'; 1) Translate 3 units right, 4 units down. 8) Reflect triangle ABC over the line $$y = -1$$. Label the image A'B'C'. 3) Dilate by a scale factor of $$\frac{1}{2}$$, centered at origin.”

• In 8.G.B.8, Seeking Safe Harbor, students apply the Pythagorean Theorem to determine the distance between the take off and landing points of a helicopter shown on a map. In Practice Printable, Question 4 states, “A medical helicopter flies about 150 miles per hour. Use the map to determine approximately how long it will take the helicopter to reach the hospital from the crash site.”

Examples of multiple aspects of rigor being engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study include:

• In 8.EE.A.4, The Great Discovery, students use procedural skills and understanding of scientific notation to solve real-world problems. In the Practice Printable, Question 3 states, “A wasp is $$1.39⋅10^{-4}$$ meters long. A pygmy rabbit is $$2.4⋅10^{-1}$$ meters long. How
 much longer is the pygmy rabbit than the wasp?” Question 4, “The speed of light is $$3⋅10^6$$ kilometers per second. How far does light travel in an hour?”

• In 8.EE.C.7b, Petunia’s Pickle Problem, students use procedural skills to engage in application problems. In the lesson narrative, “During Petunia’s Pickle Problem, Petunia and her mother, Rosa, make and sell pickles, at times even traveling to sell their pickles. Rosa tells Petunia that it’s important in any business to at least break even, or bring in as much money as they spent. Rosa then asks Petunia to help her figure out how many jars of pickles they must sell to break even. The data provided is Petunia’s notebook, detailing expense and income information, along with the break-even equation she has written.” The resolution video teaches the procedure which is practiced throughout the lesson. Students apply their skill to real-world problems in the Practice Printable, Question 8, “Omarion and Savannah are both saving money for their summer trip. Omarion started with $130 and puts in $10 every week. Savannah started with $55 and puts in $35 every week. a) Write and solve an equation that will determine the number of weeks ($$w$$) when Savannah and Omarion have the same amount of savings. b) At that point, how much savings will they each have?”

The materials reviewed for Core Curriculum by MidSchoolMath 5-8 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.

Examples where MP1 (Make sense of problems and persevere in solving them) is connected to grade-level content include:

• In 8.G.A.3, Tile Transformation, Detailed Lesson Plan, In Immersion and Data & Computation, students will use their knowledge of transformations to identify final coordinates for a figure. They will make conjectures about which transformations will give the desired results to Miss Cartesia. Students will devise a plan which will involve a bit of experimentation and revision as they attempt to manipulate the original figure. This is a great lesson to apply Jo Boaler’s tip to consider your teacher messages in influencing student ability to make sense of problems and persevere.”

• In 8.EE.C.8b, Mars Rocks!, Lesson Plan Overview, Applying Standards for Mathematical Practice, “In Immersion, students engage in a brief discussion where they begin to make meaning of the problem and of what is being asked, allowing them to formulate an initial solution pathway. Upon receiving additional information in Data & Computation, students use what they know about graphs, coordinates, lines and equations to find their own entry point to the problem, which may very well change course throughout their work as they find strategies that aren't effective.”

• In 8.NS.A.2, Lesson Plan Overview, Applying Standards for Mathematical Practice, “Treasure Hunt provides a rare opportunity for students to see an irrational number (square root of 2) in a real-world context, which creates a sense of meaning and perseverance for students. Protocols that include discourse between students enhance student understanding of the problem. In particular, in Immersion, students are prompted to draw a visual representation of what the problem is asking in context and are asked to explain their drawing to a partner. Further, in Data & Computation, students analyze each other’s work, tasked with finding places of agreement, including the best approximation for $$\sqrt{2}$$.”

Examples where MP2 (Reason abstractly and quantitatively) is connected to grade-level content include:

• In 8.EE.A.2, Ship Shape, Lesson Plan Overview, Applying Standards for Mathematical Practice, “In Resolution, students are tasked with creating a drawing of a shipping container of any dimensions that would fit cubic boxes with a volume 729 cubic feet without wasting space, and tasked with determining how many boxes of that size it would hold. Many volume problems that ask how much of something will fit inside a container are typically solved by dividing the volume of the container by the volume of what will go inside. However, problems like the one in Ship Shape, where the dimensions of the container constrain the number of boxes that physically fit (no matter the volume), require more attention. Abstract calculations must be contextualized to come to an accurate and practical solution.”

• In 8.F.A.1, Flight Functions, Lesson Plan Overview, “MP2: Reason abstractly and quantitatively. During Data & Computation, students recognize that a function is a special relationship between two variables and will distinguish between functions and non-functions. Students understand abstractly that a function is a set of input values each with exactly one output value and that it can be represented in multiple ways.”

• In 8.EE.C.7b, Petunia’s Pickle Problem, Lesson Plan Overview, Applying Standards for Mathematical Practice, “During Immersion and Data & Computation, students comprehend the story in context, but must calculate the solution after simplifying the equation and then recontextualize the solution as the number of jars of pickles needed to break even.”

The materials reviewed for Core Curriculum by MidSchoolMath 5-8 Grade 8 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 include 10 protocols to support Mathematical Practices. Several of these protocols engage students in constructing arguments and analyzing the arguments of others. When they are included in a lesson, the materials provide directions or prompts for the teacher to support engaging students in MP3. These include: 

• “Lawyer Up! (12-17 min): When a task has the classroom divided between two answers or ideas, divide students into groups of four with two attorneys on each side. Tell each attorney team to prepare a defense for their ‘case’ (≈ 4 min). Instruct students to present their argument. Each attorney is given one minute to present their view, alternating sides (≈ 4 min). Together, the attorneys must decide which case is more defendable (≈ 1 min). Tally results of each group to determine which case wins (≈ 1-2 min). Complete the protocol with a ‘popcorn-style’ case summary (≈ 2-3 min).”

• “Math Circles (15-28 min): Prior to class, create 5 to 7 engaging questions at grade level, place on different [sic] table-tops. For example, Why does a circle have 360 degrees and a triangle 180 degrees? Assign groups to take turns at each table to discuss concepts (≈ 3-4 min each table).”

• “Quick Write (8-10 min): After showing an Immersion video, provide students with a unique prompt, such as: ‘I believe that the store owner should…’, or ‘The person on Mars should make the decision to…’ and include the prompt, ‘because…’ with blank space above and below. Quick writes are excellent for new concepts (≈ 8-10 min).”

• “Sketch It! (11-13 min): Tell students to draw a picture that includes both the story and math components that create a visual representation of the math concept (≈ 5-7 min). Choose two students with varying approaches to present their work (≈ 1 min each) to the class (via MidSchoolMath software platform or other method) and prepare the entire class to discuss the advantages of each model (≈ 5 min).”

The materials include examples of prompting students to construct viable arguments and critique the arguments of others.

• In 8.EE.B.5, Space Race, Practice Printable, Introduction Problem, “Which ship will get to Candoran first? Commander K-8 has three teams entering the annual race to Candoran. The teams have been practicing all week and have been keeping track of their practice data. Use the practice data to help Commander K-8 predict who will win the race. Explain your reasoning.” Data for each team is provided with a different representation: equation, table, graph. Practice Printable Question 4, “Compare the proportional relationships (given Bob’s data in a table and Liv’s data in a graph). Which of the two stores charges the least for greeting cards? Explain your reasoning.”

• In 8.EE.C.8a, Show Me the Money, Practice Printable, Introduction Problem, “How could Bailey use a graph to explain Cage’s contract offers? Football quarterback, Tom Cage and his agent, Bailey Simpson, are reviewing recent offers that have come in from two teams, the Kingsboro Kangaroos and the Hartford Hedgehogs. Cage is having difficulty understanding the equations in the contracts, so he asks Bailey to explain to him using a graph. Use the equations to graph the line for each team contract. Then explain what this means for Cage.”

• In 8.SP.A.2, Escape from Mars, Practice Printable, Introduction Problem, “What battery should she take? Team Delta Geologist Kim O’Hara is trying to make it back to Earth. She believes there is an escape pod 59 kilometers away from her base. To get there she must take the exploration vehicle. The battery can only be charged at her home base, and she’s unsure if it will last long enough to make it to the escape pod. Luckily, she found two batteries, but, sadly, she can only take one. She runs tests on both batteries and creates a scatterplot for each, showing the remaining battery life after traveling various distances. For each scatterplot, draw a line of best fit. Then decide which battery O’Hara should take and explain your reasoning.”

• In 8.G.A.2, Knee Replacement, Practice Printable, Question 1, “A student claims that any two shapes are congruent if one can be formed from the other by a series of rotations or reflections. Is the student correct? Explain your answer.” Question 2 states, “A transformation is made to triangle ABC to form triangle DEF (not shown). Then another transformation is made to triangle DEF to form triangle GHJ. Describe these transformations. What is the relationship between triangles ABC and GHJ? Explain your answer.” Question 3 states, “Prove that ABCD is congruent to JKLM. Describe the transformations that would be made to ABCD to form JKLM.” Question 4 states, “Are the shapes congruent? How could you use transformations to prove your answer?”

• In 8.F.A.1, Flight Functions, Practice Printable, Question 1, “Determine whether each relationship represents a function and explain why or why not.” Question 3 states, “Determine whether each of the following could be a graph of a function, and explain why or why not: a) A line that gradually increases in a diagonal manner; b) A horizontal line; c) A circle; d) A vertical line.”

The materials provide guidance for teachers on how to engage students with MP3. In several lessons, the Detailed Lesson Plan identifies MP3 and provides prompts that support teachers in engaging students with MP3. Examples include::

• In 8.SP.A.1, Cholera Outbreak!, teachers apply Jo Boaler’s mathematical practices tip for MP3 by asking: “How do you see this problem? How do you think about it? Instructions note to follow this by having students re-state each other’s reasoning in different ways.”

• In 8.G.1a-c, Artemis Transforms, the final step in Data and Computation, has teachers pair students, present their case to each other, and justify their reasoning. Prompts include: “Is the other team’s argument logical? What evidence did they provide? What would have made their argument stronger?”

The materials reviewed for Core Curriculum by MidSchoolMath 5-8 Grade 8 meet expectations for supporting the intentional development of MP4 and MP5 for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Examples of the intentional development of MP4 to meet its full intent in connection to grade-level content include:

• In 8.EE.C.7a, The Business Guru in YOU!, the Detailed Lesson Plan states, “MP4: Model with mathematics. The task for The Business Guru in YOU! is fundamentally a practice in modeling with mathematics. Too often, modeling is considered in the more limited sense where a real-life context is modeled with an equation. In this lesson the task is reversed, where students create a ‘real-life’ situation to match an equation. During Data & Computation, students are tasked with creating the story behind each of these businesses. Protocols support their practice to attempt to create a business story that could be represented by an equation. In Resolution, students are tasked with all aspects of the problem: to create an equation, create a business story it could represent, and transform the equation into a simpler form. Protocols support students seeing how others represented their equations.”

• In 8.EE.C.8a, Show Me the Money, Lesson Plan Overview, “MP4: Model with Mathematics. During Immersion and Data & Computation, students analyze equations formed from real-world contexts and connect them with graphical representations. From the unstructured problem, they develop a system of linear equations, and explain how this connects to the real-world context of the problem.”

• In 8.G.C.9, Dawn of Anesthesia, the Detailed Lesson Plan states, “MP4: Model with mathematics. In Immersion, students determine what they need to know in an unstructured problem and create strategies based on assumptions using Jo Boaler’s recommendation of formalizing a model on a poster to help clarify thinking. Because this is a difficult problem to model and because it has complex elements, such as the spherical bottle, students may initially oversimplify at this stage. In Data & Computation, students have the opportunity to improve their assumptions and model. In Resolution, teachers immediately reinforce student learning with a special modeling exercise from Dan Meyer’s Meatballs problem. Students engage in modeling to determine how many spherical meatballs fit in a cylindrical pot.”

Examples of the intentional development of MP5 to meet its full intent in connection to grade-level content include:

• In 8.G.A.4, Dakota Jones and the Hall of Records, the Detailed Lesson Plan states, “MP5: Use appropriate tools strategically. During Data & Computation, students are instructed to consider the tools needed to come to a conclusion about the Hall of Records. The specific tools used vary from student to student, including but not limited to paper & pencil, trace paper, graph paper, rulers, chart paper, computational software, graphics editing software, presentation and animation software. Students will need various tools to show that similar figures can be created through transformations and used as models to support their claim.”

• In 8.SP.A.3, The Slope of Sprouts, the Detailed Lesson Plan states, “MP5: Use appropriate tools strategically. In Immersion, students are asked to consider what tools might be helpful to solve this problem, keeping in mind what O'Hara is trying to find out about her sprouts. In Data & Computation, students have the opportunity to use the tools they find useful as they interact with the given scatter plot.”

The materials reviewed for Core Curriculum by MidSchoolMath 5-8 Grade 8 meet expectations for supporting the intentional development of MP6: Attend to precision for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The materials use precise and accurate terminology and definitions when describing mathematics, and the materials provide instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Examples include:

• Each Detailed Lesson Plan provides teachers with a list of vocabulary words and definitions that correspond to the language of the standard that is attached to the lesson; usually specific to content, but sometimes more general. For example, 8.G.2 states “Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent two-dimensional figures, describe a sequence that exhibits the congruence between them.” The vocabulary provided to the teacher in 8.G.A.2, Knee Replacement is, “Congruent: Equal in shape and size.” 

• The vocabulary provided for the teacher is highlighted in red in the student materials on the Practice Printable.

• Each Detailed Lesson Plan prompts teachers to “Look for opportunities to clarify vocabulary” while students work on the Immersion problem which includes, “As students explain their reasoning to you and to classmates, look for opportunities to clarify their vocabulary. Allow students to ‘get their idea out’ using their own language but when possible, make clarifying statements using precise vocabulary to say the same thing. This allows students to hear the vocabulary in context, which is among the strongest methods for learning vocabulary.” 

• Each Detailed Lesson Plan includes this reminder, “Vocabulary Protocols: In your math classroom, make a Word Wall to hang and refer to vocabulary words throughout the lesson. As a whole-class exercise, create a visual representation and definition once students have had time to use their new words throughout a lesson. In the Practice Printable, remind students that key vocabulary words are highlighted. Definitions are available at the upper right in their student account. In the Student Reflection, the rubric lists the key vocabulary words for the lesson. Students are required to use these vocabulary words to explain, in narrative form, the math experienced in this lesson. During ‘Gallery Walks,’ vocabulary can be a focus of the ‘I Wonder..., I Notice…’ protocol.”

• Each lesson includes student reflection. Students are provided with a list of vocabulary words from the lesson to help them include appropriate math vocabulary in the reflection. The rubric for the reflection includes, “I clearly described how the math is used in the story and used appropriate math vocabulary.” 

• Vocabulary for students is provided in the Glossary in the student workbook. “This glossary contains terms and definitions used in MidSchoolMath Comprehensive Curriculum, including 5th to 8th grades.” 

• The Teacher Instruction portion of each detailed lesson plan begins with, “Here are examples of statements you might make to the class:” which often, though not always, includes the vocabulary used in context. For example, the vocabulary provided for 8.F.A.3, Le Monsieur Chef is “Linear Function,” and “Non-linear Function.” The sample statements provided are, “Remember that a function is defined by the input value resulting in only one output value. In Le Monsieur Chef, we were asked to determine if each equation was a function and if it was linear or non-linear. The chef helped us see that linear equations can all be written in the slope-intercept form or y = mx + b, where b is the y-intercept and starting value and m is the slope and rate of change.”

The materials reviewed for Core Curriculum by MidSchoolMath 5-8 Grade 8 meet expectations for supporting the intentional development of MP7 and MP8 for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Examples of the intentional development of MP7 to meet its full intent in connection to grade-level content include:

• In 8.F.A.3, Le Monsieur Chef, the Detailed Lesson Plan states, “On Day 1, during both the Immersion and Data & Computation phases, students will recognize that not all equations are linear. They will recognize that all linear equations can be written in the same form.” Optional teacher prompts include, “Does the equation represent a function? What does linear mean? What does non-linear mean? Can the equation be written in slope-intercept form? Is the rate of change constant? How do you know? Are there exponents in the equation? What might the graph look like?”

• In 8.G.A.5, Puppy Parallels, Lesson Plan Overview, “MP7: Look for and make use of structure. During Immersion and Data & Computation, students recognize that the structure of two lines cut by a transversal creates many congruent angles. They analyze the structure of various diagrams and utilize prior knowledge surrounding supplementary angles. During Resolution, students see that the structure of parallel lines with a transversal lends itself to translation to see additional relationships among the angles.”

Examples of the intentional development of MP8 to meet its full intent in connection to grade-level content include:

• In 8.EE.A.1, The Big Shrink, the Detailed Lesson Plan states, “MP8: Look for and express regularity in repeated reasoning. After Resolution, students complete a table whereby they will experience and recognize repeated reasoning in multiplication of exponents with the same base. Afterward, they engage in a ‘Number Talk’ which directs students to explicitly notice the regularity of the pattern leading to the general Product Rule for exponents.”

• In 8.F.A.2, Happy Trails, the Detailed Lesson Plan states, “MP8: Look for and express regularity in repeated reasoning. On Day 1, during both the Immersion and Data & Computation phases, students will identify quantitative relationships provided in multiple formats. They will then discover that the rate of travel calculations for Brogan’s Trail are repeated.”

• In 8.NS.A.1, Warp Speed, Detailed Lesson Plan, “MP8: Look for and express regularity in repeated reasoning. This lesson provides an opportunity for students to notice repeated calculations for general methods and shortcuts. In Data & Computation and Teacher Instruction, students look for patterns and observe how using powers of 10 can help to algebraically convert decimal expansions to their fractional equivalents, paying particular attention to how many digits are repeating in the decimal expansion.”