High School - Gateway 3

The materials reviewed for Math Nation AGA meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Examples of where and how the materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials include:

• Course Overview: A Course Overview (Unit 0) is found at the beginning of each course.  Within each Course Overview, there is a Course Narrative, which contains a summary of the mathematical content contained in each course. The Course Overview contains two main components: Course Guide and Modeling Prompts. The Course Guide contains the following sections: Introduction, About These Materials, How to Use These Materials, Assessment Overview, Scope and Sequence, Required Materials, and Cool Down Guidance. Each of these sections contains specific guidance for teachers on implementing lesson instruction. For example, in the “About These Materials” section, teachers can find an outline of and detailed information about the components of a typical lesson, including Warm-Up, Classroom Activities, Lesson Synthesis, and Cool Down. The “How to Use These Materials” section contains guidance about utilizing instructional routines and digital routines, which include applications of technology, and in the Scope and Sequence section, teachers will find a Pacing Guide, which contains time estimates for coverage of each of the units. The Modeling Prompts component is divided into two sections, Modeling Prompts- Overview and Mathematical Modeling Prompts. Within these sections, teachers can find guidance on introducing the modeling cycle to students, selection of modeling prompts related to units, and lessons contained in the course.

• Teacher Edition: There is a Teacher Edition section for each unit that contains a unit introduction, unit assessments, and unit-level downloads. The Unit Introduction contains a summary of the mathematical content to be found in the unit. The Assessment component contains downloads for multiple types of assessment (Check Your Readiness, Mid-Unit, and End-of-Unit Assessment). Unit Level Downloads include: Teacher Guide, Assessments, Unit at a Glance, Blackline Masters, Lesson Plans, and Teacher Presentation Materials, all of which provide support for teacher planning. Each lesson has a Teacher Edition component that contains guidance for Lesson Preparation, Supports, Cool-down Guidance, and a Lesson Narrative. The Lesson Preparation component includes a Teacher Prep Video, Learning Goal(s), Required Material(s), and Teacher Guide downloads. Cool-down Guidance provides teachers with guidance on what to look for or emphasize over the next several lessons to support students in advancing their current understanding. The Lesson Narrative provides specific guidance about how students can work with the lesson activities.

• Teacher Guide: Within each Teacher Edition lesson component, teachers can find a Teacher Guide that contains lesson learning goals and targets, a lesson narrative, and specific guidance for implementing each of the lesson activities. The Lesson Narrative contains the purpose of the lesson, standards and mathematical practices alignments, specific instructional routines, and required materials related to the lesson. Teachers are given guidance for implementing these routines as a way of introducing students to the learning targets. There is also teacher guidance for launching lesson activities, such as suggestions for grouping students, working with a partner, or whole group discussion. The planning section identifies possible student errors and misconceptions that could occur. There is also guidance on how to support English Language Learners and Students with Disabilities.

Evidence for materials including sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives to engage and guide student mathematical learning include:

• Algebra 1, Unit 5, Lesson 11, Teacher Guide, 5.11.2 Exploration Activity, “In this activity, students examine the successive heights that a tennis ball reaches after several bounces on a hard surface and consider how to model the relationship between the number of bounces and the height of the rebound. To do so, they need to determine the growth factor of successive bounce heights. Because some data is provided here, students engage in only some aspects of mathematical modeling. To engage students in the full modeling cycle that includes data gathering, consider asking students to measure the bounce heights of a ball, as suggested in the next optional activity. Real-world data is often messy and that is the case for the data provided here…Encourage those who do not think an exponential model is appropriate to look for an exponential model that best fits the given data. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).” 

• Geometry, Unit 2, Lesson 6, Teacher Guide, 2.6.2 Exploration Activity, “Activity Synthesis The goal of this discussion is to continue to emphasize that proofs using transformations are generalized statements that work for all triangles that match the given criteria, rather than just one specific drawing. Select students whose triangles require translation and rotation but not reflection to share their drawings and the steps in their transformations. Record a proof these triangles are congruent…Point out the concluding statement. Add this to the display of sentence frames for proofs. This display should be posted in the classroom for the remaining lessons within this unit. You will add to the display throughout the unit. An example template is provided in the blackline masters for this lesson.”

• Algebra 2, Unit 3, Lesson 6, Teacher Guide, 3.6.3 Exploration Activity, “Anticipated Misconceptions, Since students usually see x-values on the horizontal axis and y-values on the vertical, they may look for a or s values on the wrong axis. Encourage students to annotate the graph by drawing horizontal or vertical lines that will intersect the curves at the point that represents the solution, or using some other method that is helpful for them. For example, when solving s^2=25, students can draw a line representing t=25 and see where it hits the graph of t=s^2, since these points represent the solutions.”

The materials reviewed for Math Nation AGA meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

The Course Guide, About These Materials sections, states the following note about standards alignment, “There are three kinds of alignments to standards in these materials: building on, addressing, and building towards. Oftentimes a particular standard requires weeks, months, or years to achieve, in many cases building on work in prior grade-levels. When an activity reflects the work of prior grades but is being used to bridge to a grade-level standard, alignments are indicated as ‘building on.’ When an activity is laying the foundation for a grade-level standard but has not yet reached the level of the standard, the alignment is indicated as ‘building towards.’ When a task is focused on the grade-level work, the alignment is indicated as ‘addressing.’” All lessons in the materials have this correlation information. Examples include:

• Algebra 1, Unit 3, Lesson 5, Teacher Guide, Alignments, Building on 8.SP.2; Addressing S-ID.6, S-ID.7; Building Towards S-ID.6, S-ID.6b.

• Geometry, Unit 5, Lesson 10, Teacher Guide, Alignments, Building on G-GMD.4; Addressing G-GMD.1, G-GMD.4, G-MG.1; Building Towards G-GMD.1.

• Algebra 2, Unit 6, Lesson 9, Teacher Guide, Alignments, Building on F-BF.1b; Addressing F-IF.7, F-TF.A; Building Towards F-IF.7.

Explanations of the role of the specific course-level mathematics in the context of the series can be found throughout the materials including but not limited to the Course Guide, Scope and Sequence section, the Course Overview, Unit Introduction, Lesson Narrative, and Full Lesson Plan. Examples include:

• Algebra 1, Unit 2, Lesson 6, Lesson Narrative, “In middle school, students learned that two expressions are equivalent if they have the same value for all values of the variables in the expressions. They wrote equivalent expressions by applying properties of operations, combining like terms, or rewriting parts of an expression. In this lesson, students learn that equivalent equations are equations with the exact same solutions…The emphasis of this lesson is on equations in one variable. Students will have many opportunities to study equivalent equations in two variables in future lessons.”

• Geometry, Unit 3, Unit Introduction, “Before starting this unit, students are familiar with dilations and similarity from work in grade 8. They have experimentally confirmed properties of dilations, and informally justified that figures are similar by finding a sequence of rigid motions and dilations that takes one figure onto the other…In a previous unit, students used rigid transformations to justify the triangle congruence theorems of Euclidean geometry: Side-Side-Side Triangle Congruence Theorem, Side-Angle-Side Triangle Congruence Theorem, and Angle-Side-Angle Triangle Congruence Theorem. In this unit, students use dilations and rigid transformations to justify triangles that are similar... This unit previews many of the important concepts that students rely on to make sense of trigonometry in later units. The latter part of the unit focuses on similar right triangles. In addition, students are introduced to some of the applications of right triangles that they will explore in more depth in the trigonometry unit, such as finding the heights of objects through indirect measurement.”

• Algebra 2, Unit 5, Unit Introduction, “Prior to this unit, students have worked with a variety of function types, such as polynomial, radical, and exponential. The purpose of this unit is for students to consider functions as a whole and understand how they can be transformed to fit the needs of a situation, which is an aspect of modeling with mathematics (MP4). An important takeaway of the unit is that we can transform functions in a predictable manner using translations, reflections, scale factors, and by combining multiple functions…The unit begins with students informally describing transformations of graphs, eliciting their prior knowledge and establishing language that will be refined throughout the unit…In a future unit, students use their knowledge of transformations to transform trigonometric functions to model a variety of periodic situations. By saving the introduction of trigonometric functions until after a study of transformations, students have the opportunity to revisit transformations from a new perspective which reinforces the idea that all functions, even periodic ones, behave the same way with respect to translations, reflections, and scale factors.”

The materials reviewed for Math Nation AGA meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. Instructional approaches of the program and identification of the research-based strategies can be found throughout the materials, but particularly in the Course Guide, About These Materials, and How to Use These Materials as well as the Mathematical Modeling Prompts- Overview sections. 

The About These Materials section states the following about the instructional approach of the program, “What is a Problem Based Curriculum? In a problem-based curriculum, students work on carefully crafted and sequenced mathematics problems during most of the instructional time. Teachers help students understand the problems and guide discussions to ensure the mathematical takeaways are clear to all. Some concepts and procedures follow from definitions and prior knowledge so students can, with appropriately constructed problems, see this for themselves. In the process, they explain their ideas and reasoning and learn to communicate mathematical ideas. The goal is to give students just enough background and tools to solve initial problems successfully, and then set them to increasingly sophisticated problems as their expertise increases. However, not all mathematical knowledge can be discovered, so direct instruction is sometimes appropriate. A problem-based approach may require a significant realignment of the way math class is understood by all stakeholders in a student's education. Families, students, teachers, and administrators may need support making this shift. The materials are designed with these supports in mind. Family materials are included for each unit and assist with the big mathematical ideas within the unit. Lesson and activity narratives, Anticipated Misconceptions, and instructional supports provide professional learning opportunities for teachers and leaders. The value of a problem-based approach is that students spend most of their time in math class doing mathematics: making sense of problems, estimating, trying different approaches, selecting and using appropriate tools, evaluating the reasonableness of their answers, interpreting the significance of their answers, noticing patterns and making generalizations, explaining their reasoning verbally and in writing, listening to the reasoning of others, and building their understanding. Mathematics is not a spectator sport.”

Examples of materials including and referencing research-based strategies include:

• “The Five Practices Selected activities are structured using Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014). These activities include a presentation of a task or problem (may be print or other media) where student approaches are anticipated ahead of time. Students first engage in independent think-time followed by partner or small-group work on the problem…”

• “Instructional Routines … Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team…”

• “Mathematical Modeling Prompts - Overview Mathematics is a tool for understanding the world better and making decisions. School mathematics instruction often neglects giving students opportunities to understand this, and reduces mathematics to disconnected rules for moving symbols around on paper. Mathematical modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions (NGA 2010). This mathematics will remain important beyond high school in students' lives and education after high school (NCEE 2013).”

Within the Modeling Prompts, Mathematical Modeling Prompts - Overview, a References section is included.

The materials reviewed for Math Nation AGA meet expectations for providing a comprehensive list of supplies needed to support instructional activities. Comprehensive lists of supplies needed to support the instructional activities can be found in Course Guides (Required Material(s)), Teacher Editions, in each lesson under Lesson Preparation (Required Materials), and in Teacher Guides for specific lessons. Examples include:

• Algebra 1, Unit 4, Lesson 16, Lesson Preparation, Required Material(s): “Pre-printed slips, cuts from copies of the blackline master”

• Geometry, Unit 3, Lesson 1, Lesson Preparation, Required Material(s): “Geometry toolkits (HS), Protractors, Rulers marked with centimeters”

• Algebra 2, Unit 2, Lesson 1, Lesson Preparation, Required Material(s): “Blank paper, Graph technology, Rulers, Scissors, Tape”

The materials reviewed for Math Nation AGA meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Student sample responses are provided for all assessments. Rubrics are provided for scoring restricted constructed response and extended response questions on the Mid-Unit Assessments and End-of-Unit Assessments. Mid-Unit Assessments and End-of-Unit Assessments include notes that provide guidance for teachers to interpret student understanding and make sense of students’ correct/incorrect responses. 

Suggestions to teachers for following-up with students are provided throughout the materials via the Check-Your-Readiness, Mid-Unit, and End-of-Unit Teacher Guides, and each lesson provides a Cool-down Guidance that details how to support student learning.

Examples of the assessment system providing multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance include:

• Course Guide, Assessments Overview states the following: “Rubrics for Evaluating Students Answers Restricted constructed response and extended response items have rubrics that can be used to evaluate the level of student responses. 

  • Restricted Constructed Response

    • Tier 1 response: Work is complete and correct.

    • Tier 2 response: Work shows General conceptual understanding and mastery, with some errors.

    • Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Two or more error types from Tier 2 response can be given as the reason for a Tier 3 response instead of listing combinations.

  • Extended Response

    • Tier 1 response: Work is complete and correct, with complete explanation or justification. 

    • Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. 

    • Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. 

    • Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.”

• Algebra 2, Unit 7, End-of-Unit Assessment, Question 6, “A scientist is worried about a new disease being spread by mosquitoes in an area. The scientist captures a random sample of 137 mosquitoes from the area and tests them to find out whether they carry the disease. In this sample, 22 of the mosquitoes were carrying the disease. Using this information, the scientist uses 100 simulations of additional samples with the same proportion of disease-carrying mosquitoes from the sample to determine that the standard deviation for the proportion of mosquitoes carrying the disease is approximately 0.029. 1. Estimate the proportion of mosquitoes in the area that carry the disease, and provide a margin of error. Explain or show your reasoning. 2. Why did the scientist run the simulations to find additional possible proportions of mosquitoes in the area that carry the disease? Solution 1. The proportion of mosquitoes in the area that carry the disease is approximately 0.16(\frac{22}{137}\approx 0.16) with a margin of error of 0.058(0.029 \cdot 2=0.058). 2. Sample: With the additional simulated proportions, a sense of the variability expected among samples from the population can be used to estimate a margin of error for the sample proportion. Minimal Tier 1 response: Work is complete and correct. Sample: 1. \frac{22}{137}=.16 The margin of error is 2 \cdot .029=0.058. 2. They needed to know the standard deviation to get the margin of error. Tier 2 response: Work shows general conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Sample errors: minor calculation mistakes in the proportion of mosquitoes or margin of error; formula used to calculate margin of error is incorrect but based on the variability of the sample proportions (for instance, using standard deviation rather than twice the standard deviation); correct answers to part a with no work shown; explanation in part b does not connect to margin of error (in name or concept) but does refer to variability or standard deviation in some way.  Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: errors in part a are more serious than Tier 2 types; response to part b does not refer to variability or standard deviation in some way.”

Examples of the assessment system providing multiple opportunities to determine students' learning and suggestions to teachers for following-up with students include:

• Course Guide, Cool-Down Guidance states the following: “Each cool-down is placed into one of three support levels: 1. More chances. This is often associated with lessons that are exploring or playing with a new concept. Unfinished learning for these cool-downs is expected and no modifications need to be made for upcoming lessons. 2. Points to emphasize. For cool-downs on this level of support, no major accommodations should be made, but it will help to emphasize related content in upcoming lessons. Monitor the student who have unfinished learning throughout the next few lessons and work with them to become more familiar with parts of the lesson associated with this cool-down. Perhaps add a few minutes to the following class to address related practice problems, directly discuss the cool-down in the launch or synthesis of the warm-up of the next lesson, or strategically select students to share their thinking about related topics in the upcoming lessons. 3. Press pause. This advises a small pause before continuing movement through the curriculum to make sure the base is strong. Often, upcoming lessons rely on student understanding of the ideas from this cool-down, so some time should be used to address any unfinished learning before moving on to the next lesson.”

• Algebra 1, Unit 1, Lesson 5, Cool-down Guidance, “Support Level 3. Press Pause. Notes Use the results from the Check Your Readiness Assessment to anticipate student struggle with MAD. Consider using Algebra 1 Supports Lesson 5 before this lesson if students need substantial support calculating MAD. Students will have more opportunities with IQR and the concept of variability.” 

• Geometry, Unit 1, Check-Your-Readiness, Question 8, “Students should be comfortable drawing transformations on the grid. During this unit, they will extend that understanding to the plane without the structure of a grid. If students miss the first part of the question, then you need to pay particular attention to the isometric grid translations in lesson 10. If they miss the second part of the question, then you need to pay particular attention to the isometric grid rotations in lesson 13.”

The materials reviewed for Math Nation AGA meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of the course-level standards and practices across the series.

All assessments regularly demonstrate the full intent of course-level content and practice standards through a variety of item types such as multiple choice, short answer, extended response prompts, graphing, mistake analysis, and constructed response items. Assessments are to be downloaded as Word documents or PDFs and designed to be printed and administered in the classroom. Examples Include:  

• Algebra 1, Unit 2, Mid-Unit Assessment, Question 3, demonstrates the full intent of A-REI.10. “Tickets to the zoo cost $12 for adults and $8 for children. The school has a budget of $240 for the field trip. An equation representing the budget for the trip is 240 = 12x + 8y. Here is a graph of this equation. (graph displayed in the problem).  Select all the true statements. A. If no adult chaperones were needed, 30 children could go to the zoo. B.  If ten children go to the zoo, then 15 adults can go. C. If four more adults go to the zoo, that means there will be room for six fewer children. D. If two more children go to the zoo, that means there will be room for three fewer adults. E. If 16 adults go to the zoo, then 6 children can go.”

• Geometry, Unit 1, End-of-Unit Assessment, Question 5, demonstrates the full intent of G-CO.9, MP1, and MP4. “Lines x and y are parallel. Write an equation that represents the relationship between b and e. Explain how you know this equation is always true.” A diagram is shown of a transversal crossing parallel lines, with multiple angles labeled.

• Algebra 2, Unit 5, Lesson 5, Cool Down, demonstrates the full intent of F-BF.3. “Let h be a function where y = h(x). 1. What is the value of a if h is an even function? 2. What is the value of a if h is an odd function?” A table is provided with the columns x and y with the corresponding entries (-4, 0, 4) and (a, 0, \frac{1}{2}).

The materials reviewed for Math Nation AGA meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning series mathematics.

Throughout the materials, strategies, supports, and resources for students in special populations can be found in the Teacher Guides and Student Editions. In the Teacher Guide, teachers can find guidance for supporting students in special populations within various lesson activities. These supports are highlighted in green and labeled “Access for Students with Disabilities”. Additionally, differentiated videos explaining course content - varying from review to in-depth levels of explanation - are resources available for each lesson to support students.

Examples of where and how materials regularly provide strategies, supports, and resources for students in special populations to support their regular and active participation in learning series mathematics include:

• Algebra 1, Unit 5, Lesson 1, Teacher Guide, 5.1.2 Exploration Activity, “Access for Students with Disabilities Representation: Internalize Comprehension. Represent the same information through different modalities by using diagrams. Encourage students to sketch diagrams that show how the amount of money grows over the first few days. Students may benefit from this visual as they transition to the use of a table or other representation to track growth. Supports accessibility for: Conceptual processing; Visual-spatial processing.”

• Geometry, Unit 4, Lesson 7, Teacher Guide, 4.7.3 Exploration Activity, “Access for Students with Disabilities Action and Expression: Internalize Executive Functions. Provide students with a four-column table to organize. Use these column headings: angle, adjacent side, opposite side, and hypotenuse. The table will provide visual support for students to identify ratios. Supports accessibility for: Language; Organization.”  

• Algebra 2, Unit 5, Lesson 7, Teacher Guide, 5.7.2 Exploration Activity, “Access for Students with Disabilities Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. During the launch take time to review terms that students will need to access for this activity. Invite students to suggest language or diagrams to include that will support their understanding of vertical translations, horizontal translations, and the vertex. Include equations and graphs that demonstrate these translations. Supports accessibility for: Conceptual processing; Language.” 

There are several accessibility options (accessed via the wrench icon in lower left hand corner of the screen) available to students to help naviagate the materials. Examples include:

• Tools Menu allows students to change the language and access a Demos Scientific and a Desmos Graphing Calculator.

• Accessibility Menu allows students to change the language, page zoom, font style, background and font color, and enable/disable the following features: text highlighter, notes, and screen reader support. 

• UserWay allows students to adjust the following: Change contrast (4 settings), Highlight links, Enlarge text (5 settings), Adjust text spacing (4 settings), Hide images, Dyslexia Friendly, Enlarge the cursor, show a reading mask, show a reading line, Adjust line height (4 settings), Text align (5 settings), and Saturation (4 settings).

The materials reviewed for Math Nation AGA meet expectations for providing extensions and/or opportunities for students to engage with course-level mathematics at higher levels of complexity.

Course Guide, How to Use These Materials, Are You Ready For More? section states the following: “Select classroom activities include an opportunity for differentiation for students ready for more of a challenge. We think of them as the ‘mathematical dessert’ to follow the ‘mathematical entrée’ of a classroom activity. Every extension problem is made available to all students with the heading ‘Are You Ready for More?’ These problems go deeper into grade-level mathematics and often make connections between the topic at hand and other concepts. Some of these problems extend the work of the associated activity, but some of them involve work from prior grades, prior units in the course, or reflect work that is related to the K-12 curriculum but a type of problem not required by the standards. They are not routine or procedural, and they are not just ‘the same thing again but with harder numbers.’ They are intended to be used on an opt-in basis by students if they finish the main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in Are You Ready for More? problems, and it is not expected that any student works on all of them. Are You Ready for More? problems may also be good fodder for a Problem of the Week or similar structure.” If individual students were to complete these optional activities, then they might be doing more assignments than their classmates.

Examples of opportunities for advanced students to investigate grade-level mathematics content at a higher level of complexity include:

• Algebra 1, Unit 4, Lesson 5, 4.5.3 Exploration Extension: Are You Ready for More?, “Describe a different data plan that, for any amount of data used, would cost no more than one of the given plans and no less than the other given plan. Explain or show how you know this data plan would meet these requirements.” This extension follows an exploration activity in which students work with two functions.

• Geometry, Unit 1, Lesson 14, 1.14.4 Exploration Extension: Are You Ready for More?, “You constructed an equilateral triangle by rotating a given segment around one of its endpoints by a specific angle measure. An equilateral triangle is an example of a regular polygon: a polygon with all sides congruent and all interior angles congruent. Try to construct some other regular polygons with this method.”

• Algebra 2, Unit 2, Lesson 10, 2.10.4 Exploration Extension: Are You Ready for More?, “What is a possible equation of a polynomial function that has degree 5, but whose graph has exactly three horizontal intercepts and crosses the 𝑥-axis at all three intercepts? Explain why it is not possible to have a polynomial function that has degree 4 with this property.”

The materials reviewed for Math Nation AGA meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

In the Course Guide, there is a list of the Mathematics Language Routines (MLRs) that are used in the materials. The Mathematics Language Routines are instructional routines developed to support students with emerging English language proficiency. The Mathematics Language Routines included in the material are the following:  

• MLR1: Stronger and Clearer Each Time

• MLR2: Collect and Display

• MLR3: Clarify, Critique, Correct

• MLR4: Information Gap Cards

• MLR5: Co-Craft Questions

• MLR6: Three Reads

• MLR7: Compare and Connect

• MLR8: Discussion Supports

These routines are referenced under Instructional Routines in the Teacher Guide for units and lessons to assist teachers with lesson planning. The “Access for English Language Learners” section within the Teacher Guide contains explanations of how to implement the MLRs.  

Examples of where the materials provide strategies and supports for students who read, write, and/or speak in a language other than English include:

• Algebra 1, Unit 2, Lesson 10, Teacher Guide, 2.10.3 Exploration Activity, “Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to provide opportunities for students to analyze how different mathematical forms and symbols can represent different situations. Display only the problem statement without revealing the questions that follow. Invite students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions that address quantities of each type of coin. Design Principle(s): Maximize meta-awareness; Support sense-making Representation: Internalize Comprehension. Activate or supply background knowledge about generalizing a process to create an equation for a given situation. Some students may benefit by first calculating how many nickels Andre would have if there were 0, 1, 5, or 10 dimes in the jar, and then how many dimes if there were 1, 5, or 10 nickels in the jar. Invite students to use what they notice about the processes they used to create an equation. Supports accessibility for: Visual-spatial processing; Conceptual processing.“

• Geometry, Unit 3, Lesson 9, Teacher Guide, 3.9.2 Exploration Activity, “Access for English Language Learners Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the proof that triangles with two pairs of congruent corresponding angles are similar. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, ‘What is the definition of similarity?’, ‘What is the center and scale factor of the dilation?’, and ‘How do you know that triangle A’B’C’ is congruent to PQR?’ Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why triangles with two pairs of congruent corresponding angles are similar. Design Principle(s): Optimize output (for justification); Cultivate conversation.”

• Algebra 2, Unit 1, Lesson 5, Teacher Guide, 1.5.3 Exploration Activity, “Access for English Language Learners Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: ‘I agree because . . .’ or ‘I disagree because . . .’ If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. Design Principle(s): Support sense-making.”

In the Student Edition for each lesson activity, students have access to videos which contain lesson explanations in both English and Spanish.

The materials reviewed for Math Nation AGA meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Virtual and physical manipulatives support student understanding throughout the materials. Examples include:

• Algebra 1, Unit 7, Lesson 1, 7.1.2 Exploration Activity, students cut a piece of material to create a frame around a picture. “Your teacher will give you a picture that is 7 inches by 4 inches, a piece of framing material measuring 4 inches by 2.5 inches, and a pair of scissors. Cut the framing material to create a rectangular frame for the picture. The frame should have the same thickness all the way around and have no overlaps. All of the framing material should be used (with no leftover pieces). Framing material is very expensive! You get 3 copies of the framing material, in case you make mistakes and need to recut.” 

• Geometry, Unit 5, Lesson 2, 5.2.2 Exploration Activity, students use an applet to cut a three-dimensional solid to identify the cross sections. “The triangle is a cross section formed when the plane slices through the cube. 1. Sketch predictions of all the kinds of cross sections that could be created as the plane moves through the cube. 2. The 3 red points control the movement of the plane. Click on them to move them up and down or side to side. You will see one of these movement arrows appear. Sketch any new cross sections you find after slicing.”

• Algebra 2, Unit 5, Lesson 1, 5.1.2 Exploration Activity, students use an applet to compare two functions and determine which function better fits the model. “A bottle of soda water is left outside on a cold day. The scatter plot shows the temperature 𝑇, in degrees Fahrenheit, of the bottle ℎ hours after it was left outside. Here are 2 functions you can use to model the temperature as a function of time:  f(h)=45+\frac{20}{h+0.5}  g(h)=45+33(0.5)^{h+0.5} 1. Which function better fits the shape of the data? Explain your reasoning. 2. Use the applet to zoom out on the graphs. Does this change your opinion about which function fits better? 3. Where do you see the 45 in the expression for each function on the graph? 4. For the function you thought didn't fit the shape of the data as well, how would you change it to fit better?” The applet has the two functions already graphed on it and allows students to add additional functions and points to the graph.