6th Grade - Gateway 2

The instructional materials for ORIGO Stepping Stones 2.0 Grade 6 meets expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade-level.

Cluster 6.RP.A addresses understanding ratio concepts and using ratio reasoning to solve problems. Multiple modules explore a variety of real-world applications using a few mathematical representations. Opportunities exist for students to work with ratios that call for conceptual understanding and include the use of some visual representations and different strategies. For example:

• In Module 6, Lesson 8, students discuss as a whole class how to solve, “A car travels 200 miles in 4 hours. How far does it travel in 1 hour?” Students are then shown how to use tables to organize rates. In the Student Journal students find unit rates, it is suggested they use tables (6.RP.1).
• In Module 3, Lesson 2, students build pictorial representations (tape diagrams) of ratios after discussing their thinking on ratios from prior lessons (6.RP.1).
• In Module 3, Lesson 1, introduces ratios by giving students real-world examples of ratios and having students describe the relationship between the two amounts. In 3.1, “Mix and Match ratio cards” students match real-world examples of ratio to find corresponding number ratio. Students relate abstract to concrete examples (6.RP.1).
• In Module 8, Lesson 1, links part-whole ratios to fractions using connecting cubes of two different colors. Student build a character then describe the ratio of colors used to build their character. Next, students analyze the profit of 1/4 being donated to charity and what the ratio of profit to donation would be (6.RP.1).

Cluster 6.EE addresses apply and extend previous understandings of arithmetic to algebraic expressions, reason about and solve one-variable equations and inequalities and represent and analyze quantitative relationships between dependent and independent variables. Multiple modules explore a variety of real-world applications using a few mathematical representations. Opportunities exist for students to work with expressions and equations that call for conceptual understanding and include the use of some visual representations and different strategies. For example:

• In Module 2, Lesson 1, students discuss questions such as, “What is the difference between and expression and an equation?" (6.EE.1).
• In Module 7, Lesson 1, students are presented with a visual representation of 16 blocks representing $$4^2$$. Students are led through discussions such as, “what number does this picture represent?”, “what expression can we write to match?”, and “what type of number is 16?”. The student materials engage students with visual representations of squares and have students write an expression “to name each collection of tiles” (6.EE.3).
• In Module 7, Lesson 4, students are shown how to find the greatest common factor (GCF) and use the distributive property to simplify expressions. Students then work in groups using rolling algebraic cubes to create expressions and simplify them. In the Student Journal, students first determine the GCF of an expression. Then students use the distributive property to simplify expressions, however, blank number sentences are provided (6.EE.3).

The instructional materials for ORIGO Stepping Stones 2.0 Grade 6 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skills and fluencies throughout the grade-level. Opportunities to formally practice procedural skills are found throughout practice problem sets that follow the units. Practice problem sets also include opportunities to use and practice emerging fluencies in the context of solving problems. Ongoing practice is also found in Assessment Interviews, Games, and Maintaining Concepts and Skills.

The materials attend to the Grade 6 expected fluencies, multi-digit division and multi-digit decimal operations (6.NS.2); add, subtract, multiply and divide multi-digit decimals using the standard algorithm for each operation (6.NS.3); and apply and extend previous understandings of arithmetic to algebraic expressions (6.EE.A). For example:

• In Module 3, Lessons 8-12, students use multi-digit division of whole numbers and decimals (6.NS.2,3).
• In Module 5 Interview, students calculate quotients in expressions that include fractions and whole numbers (6.NS.2,3).
• In Module 2, Lesson 8, includes using <, >, or = to compare expressions that include skills such as subtracting or multiplying decimal numbers.
• In Module 2, Lessons 9-12, students solve problems using the standard algorithm for performing mathematical operations with decimals (6.NS.3).
• In Module 3, Lesson 10, students add and subtract decimals, and multiply or divide numbers as they convert units. In Module 2, Lessons 9-12, provide practice for solving problems using the standard algorithm for performing mathematical operations with decimals (6.NS.3).
• In Module 4, Lesson 2, provides practice in writing equations to match word problems. For example, “What variable can you use to represent the value?” and “What operations will we use to calculate that value?” (6.EE.A).

In addition, the instructional materials embed opportunities for students to independently practice procedural skills and fluencies:

• The Stepping Stones 2.0 overview states that every even numbered lesson includes a section called Maintaining Concepts and Skills that incorporates practice of previously learned skills from the prior grade level.
• Each module contains a summative assessment called Interviews. According to the program, “There are certain concepts and skills, such as the ability to route count fluently, that are best assessed by interviewing students.” For example, Module 9 Interview, students must demonstrate fluency of finding the mean, median, and mode of a data set.
• Some lessons provide opportunities for students to practice procedural skills during  the Step Up section of the student journal.
• Fundamentals Games contains a variety of games that students can play to develop grade level fluency skills.

The instructional materials for ORIGO Stepping Stones 2.0 Grade 6 meet expectations that the materials are designed so teachers and students spend time working with engaging applications of the mathematics.

Engaging applications include single and multi-step word problems presented in contexts in which the mathematics is applied. There are routine problems, and students also have opportunities to engage with non-routine application problems. Thinking Tasks found at the end of Modules 3, 6, 9, and 12, provide students with problem-solving opportunities that are complex and non-routine with multiple entry points.

Examples of routine application problems include, but are not limited to:

• In Module 6, Lesson 8, addresses the standard 6.RP.3, “Antonio mixes teaspoons of yellow and red paint in the ratio of 12:4. How much yellow paint will be used for 1 teaspoon of red?”
• In Module 5, Lesson 2, addresses the standard 6.NS.1, “A straw is ten-twelfths of a foot long. Nicole cuts the straw into shorter pieces that are each two-twelfths of a foot long. How many pieces did she cut?”
• In Module 7, Lesson 10, address the standard 6.EE.7, “Three friends enter a 15-mile fun run. Dwane runs the first 4 1/2 miles. Natalie runs the next 3 1/4 miles. Reece runs the rest of the distance. How far did Reece run? Let r represent the unknown distance.”
• In Module 4, Lesson 11, addresses the standard 6.EE.9, “In a math test, a student scores 5 points for each correct answer. What is a student’s score if they get 15 correct answers?” (identifying dependent and independent variables).
• In Module 9, Lesson 12, address the standard 6.G.4, “The roof of a cottage needs refurbishing. One side of the roof is 30 ft long by 12.5 ft wide. If a bundle of shingles can be purchased to cover $$24 ft^2$$ , how many bundles are needed?”.
• Maintaining Concepts and Skills includes application exercises. For example, in Module 12, Lesson 10, “Andrea wants to buy a guitar that costs $150. The music store has a 25% off sale. When she buys the guitar she is given an extra 5% off. What amount does she pay for the guitar?”
• In Module 3, Investigation 2, students work in pairs to answer, “Emily has beetles to feed her lizards. Altogether she has a total of 52 legs. How many lizards could Emily have? What is the ratio of lizards to beetles?” (6.RP.1).

Examples of non-routine application problems with connections to real-world contexts include, but are not limited to:

• In Module 3, Thinking Task, Question 1, students read points on a coordinate grid to fill in a table showing the cost of a phone bill per month. Question 3 states, “Riku investigates the call rate for Option B. It will cost $0.30 for each minute that she spends on the phone. She decides to calculate the total cost of a six-minute call. Which of these displays is most likely to show the total cost? Explain your thinking.” This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.
• In Module 6, Thinking Task 6, Question 2 states, “An architect hired by the school district draws up these plans (design shown). She decides to split the hall into three areas: stage, auditorium, and entrance. The stage needs wooden flooring and the auditorium will be carpeted. The school has a budget of $900 to spend on the wooden stage flooring. One dimension of the stage is 8m. They would like the other dimension to be somewhere between 3 and 5 meters. Given their budget, write the dimensions for the largest stage the school can afford to build. Show your thinking.” This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.
• In Module 9, Thinking Task, students find the surface area of a greenhouse with dimensions given and identify the net that matches the greenhouse design. Question 3 states, “Archie plans to build the garden bed in this picture (3ft x 6ft x 6in deep). The measurements are taken from inside the garden bed. He will need to buy the wood to build it and the soil to fill it. He has all the other tools and materials that are necessary. What is the cost of building the garden bed?” This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.
• In Module 12, Thinking Task, students are provided a collection of data in two tables. Number of vehicles that drive past the school at certain times during the day, the other set of data is a record of each vehicle’s speed. Question 1 states, “How many vehicles traveled at a speed that is equal to or greater than 25 miles per hour?” This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

The instructional materials reviewed for ORIGO Stepping Stones 2.0 Grade 6 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

All eight MPs are clearly identified throughout the materials. For example:

• The Math Practices are initially identified in the Steps portion of each module course information.
• Videos for each module can be found under the Resources tab which explains the Math Practices and Habits of Mind.
• A table is provided to show which mathematical practices are in each lessons.
• Resources states that each practice standard is, “experienced, practiced, and enhances as a result of working on meaningful problems”.
• Module Lessons tabs have a Lesson Contents overview that lists each lesson and the standards and mathematical practices in the lesson.

The MPs are used to enhance the mathematical content and are not treated separately from  content in lessons. However, there is limited guidance for teachers on the connections between the MPs and the content standards.

The instructional materials reviewed for ORIGO Stepping Stones 2.0 Grade 6 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher guidance, questions, and sentence stems for MP3 are found in the Steps portion of the lessons. In some lessons, teachers are given questions that prompt mathematical discussions and engage students to construct viable arguments, and in other lessons, teachers are provided questions and sentence stems to facilitate students in analyzing the arguments of others, and to justify their answers.

Examples where teachers are provided guidance to engage students in constructing viable arguments and/or analyzing the think of others include, but are not limited to:

• In Module 1, Lesson 7, students compare and order positive and negative numbers on a number line. Teachers encourage students to critique their reasoning using the sentence stems, “I have a different opinion, I think, I agree (disagree) because, and That makes sense, but...”.
• In Module 4, Lesson 2, students analyze four equations to determine which correctly matches a word problem presented. Teachers ask students to justify their findings, encourage students to ask clarifying questions, and to critique the reasoning students present.
• In Module 5, Lesson 6, students explore division of common fractions by common fractions with unrelated denominators. Teachers provide sentence stems to encourage critique, “I noticed the same pattern and I also noticed, I don’t think that ... shows a pattern because, and I looked at the equations in a different way.”.
• In Module 8, Lesson 5, students work independently or in pairs to calculate the percentages and solve the problems on Slide 5. Teachers encourage students to model and record their thinking and to justify their answers. If students do not agree, they are instructed to share their own thinking and justify why it is correct.