6th Grade - Gateway 2

The instructional materials for EdGems Math Grade 6 meet 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. The instructional materials include Teacher Gems and Student Gems which provide links to activities that build conceptual understanding. Explore! activities provide students the opportunity to develop conceptual understanding at the beginning of each new lesson. In addition, Exercises, Online Practice, and Gem Challenges include problems to allow students to independently demonstrate conceptual understanding. Evidence includes:

• Lesson 1.2, Ratio Tables and Graphs, Teacher Gems includes the activity, Always, Sometimes, Never. “Decide if the statement in the box is always true, sometimes true, or never true.” Students provide conceptual evidence to support statements about ratio relationships, ratio tables, equivalent ratios, and graphing ratios. (6.RP.3a,d)
• Lesson 3.1, Introducing Percents, the Explore! activity develops conceptual understanding of percents through the use of a 10 x 10 grid. Students shade the grid to represent decimals, fractions, and percents. (6.RP.3c)
• Lesson 4.2, Dividing Fractions with Models, the Explore! activity develops conceptual understanding of fraction division by having students make connections between arithmetic expressions and multiple different visual representations. Students use these connections to solve fractional division problems. (6.NS.1)
• Lesson 8.1, Input-Output Tables, the Explore! activity builds conceptual understanding of independent and dependent variables by having students read a description, underline the quantity that is independent of the other quantity, choose one of the descriptions, and create a table of values. This provides the student with the opportunity to analyze and test values within tables to determine output with a given input. (6.EE.9)

The instructional materials for EdGems Math Grade 6 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The materials include problems and questions that develop procedural skill and fluency and provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade. The materials develop procedural skills and fluencies in Student Gems, Lesson Examples, Student Exercises, and Teacher Gems. The materials provide opportunities for students to independently demonstrate procedural skills and fluencies in Proficient, Tiered, and Challenge Practice, Online Practice, Gem Challenges, and Exit Cards. Each unit provides additional practice with procedural skills in the Student Gems. Additional practice activities are specific to the standard(s) in each lesson. Included in each unit are links to: Khan Academy, IXL Practice, and Desmos Practice. Examples of developing procedural skill and fluency include:

• Lesson 2.1, Exit Card, students build fluency in adding and subtracting decimals with multiple examples; “Find the value of: 1. 3.4 – 1.9 , 2. 52.93 + 31.4” (6.NS.3).
• Lesson 2.4, Gem Challenge 1, students find the exact quotient for several problems (6.NS.2). For example, Question 1, “Divide 19,008 ÷ 32. Find the exact quotient.” Online Practice, Question 4 states, “What is the remainder on the quotient of 1,743 divided by 11?”
• In Lesson 2.4, Dividing by Multi-Digit Numbers, students develop the procedural skill of dividing multi-digit numbers with multiple problems. For example, Proficient Practice, Problem 12, “Charlie had 150 feet of wire. If he cut the wire into 12 equal pieces, how long was each piece of wire?” (6.NS.2)
• In Lesson 5.3, Variables and Expressions, Student Gems link to Khan Academy Quiz: "Write an expression to represent: The sum of ten and the quotient of a number x and 6”. Students are provided multiple practice problems to develop procedural skills. (6.EE.2a)
• Lesson 5.1, Teacher Gems, the activity, “Matho” includes nine numerical expressions that include exponents. The students evaluate the expressions to determine if their expressions match the numbers called in a bingo-like activity. (6.EE.1)
  

The instructional materials for EdGems Math Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the grade-level mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. Students engage with materials that support non-routine and routine applications of mathematics in the Explore! activities, Teacher Gems, Performance Tasks, and Rich Tasks. Some of the Student Pages and Proficient, Tiered, and Challenge Practice allow students to engage with problems including real-world contexts and present multiple opportunities for students to independently demonstrate application of grade-level mathematics. Examples include:

• Lesson 1.4, Comparing Rates, the Explore! Activity, students apply ratio and rate reasoning in real-world problems. (6.RP.3) “Kimiko has two summer job options. She has a daily babysitting job she could do where she gets paid the same hourly rate for each hour she babysits. For a 3 hour babysitting job, she earns $21. She also has the option of helping in her mother’s laundromat. Her mom pays her $64 for every 8 hours she works.” Students complete ratio tables for each job and solve problems about the situation.
• Lesson 4.3, Dividing Fractions, Student Lesson, students apply dividing fractions in real-world examples (6.NS.1). “Shiloh drives 3/4 mile to school each day. There is a stop sign every 3/8 mile. How many stop signs will she pass on her way to school?” 
• Lesson 9.3, Area of Composite Figures, Explore! activity, students apply finding area of special polygons in real-world situations. (6.G.1) “Kienan worked for a landscaping company. He was assigned to determine how much bark was needed to put in the kids’ play area at a new park. He was given the blueprint of the polygonal play area. Some dimensions were given on the drawing and others were not. Step 1: Determine the area of the entire play area. Show your method for calculating the area and show all needed dimensions, if not given.” 
• In Lesson 6.2, Solving Addition and Subtraction Equations, Proficient Practice, students independently apply solving equations with rational numbers in a real-world example, “At the market, Sydni puts a few pears in a bag then weighs the bag on a produce scale. The bag of pears weighs 4.75 pounds. She adds more pears to the bag. The new weight of the bag is 5.6 pounds. Write and solve an addition equation to find the weight of the pears added to the bag.” (6.EE.7)

The instructional materials for EdGems Math Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

All three aspects of rigor are present independently throughout the program materials. Examples include:

• In Lesson 5.1, Powers & Exponents, students build fluency in writing exponents. For example, Student Lesson, “Write the numerical expression as a power. 3 × 3 × 3 × 3 × 3", students practice this skill with multiple problems including decimal numbers and fractions. (6.EE.1)
• In Lesson 5.5, Equivalent Expressions, Example 1, students demonstrate conceptual understanding by using a visual representation of a table to compare and test expressions to determine if they are equivalent. Students are instructed to, “Show that 2x + 5 and x + 3 + x + 2 are equivalent expressions. Create a table of values to test different input values.” (6.EE.4) 
• In Lesson 7.3, Inequalities, Teacher Gems Station activity, students write inequalities to represent real-world situations and work with non-routine problems (6.EE.8). The Stations activity includes tasks that students engage with as they move from station to station. One example included is Station 1: “Write an inequality for the statement: The amount of money Jacob has in his bank account, m, is more than $300.” Station 3: “Write an inequality for the statement: The amount of time Ronnie practices guitar, g, is less than 15 hours a week.” Station A: “Describe a situation modeled by the inequality: x < 15.”

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• In Lesson 1.2, Ratio Tables and Graphs, students develop conceptual understanding within an application problem. In Question 7, students explain how to use a ratio table to find a specific value (6.RP.3a). An example included is “Kyle sells sandwiches at the deli. He sells five turkey sandwiches for every two ham sandwiches. a. Create a ratio table showing the number of turkey sandwiches sold compared to ham sandwiches sold. Include four equivalent ratios. b. Explain how you could use a ratio table to find how many ham sandwiches were sold if 45 turkey sandwiches were sold. Include the number of ham sandwiches sold in your explanation.”
• In Lesson 1.3, Rates and Unit Rates, Proficient Practice, students develop fluency in finding unit rates as they calculate unit rates in multiple situations. (6.RP.3) Examples included are Problem 13, “Keisha drove 100 miles in 2 hours. At this rate, how far will she drive in 6 hours?” Problem 14, “Jimmy paid $75 for 3 people to attend a play downtown. If it costs the same per ticket, how much will Alan pay for 10 people to attend the play next week?”
• Lesson 4.4, Multiplying and Dividing Mixed Numbers, Student Gems link to Dan Meyer 3-Act Math tasks, students develop conceptual understanding of multiplying and dividing mixed numbers within an application (6.NS.1). An example is Nana’s Lemonade where students watch a 3 part video, make predictions about the ratio of water to lemon juice, pose questions, answer questions, and compare their thinking with their peers.
• In Lesson 3.3, Percents of a Number, students develop conceptual understanding simultaneously with procedural skill to find a percent of a quantity (6.RP.3c). The Explore! activity in this lesson provides students with a diagram of a grid representing a room with pieces of furniture and guides students through the process of finding the percent of the room taken up by each object. The directions state: “Kieran used a piece of 4 by 5 grid paper to sketch the floor plan of his room. He colored the location of his bed, dresser and desk.”
• Step 1: Write the ratio of the parts shaded for each object to the total space as a fraction.
• Step 2: Convert each fraction in Step 1 to a decimal fraction with a denominator of 100.
• Step 3: What percentage of the room is taken up by each object?

The instructional materials reviewed for EdGems Math Grade 6 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade level.

All 8 MPs are identified throughout the materials. Each lesson includes a Lesson Guide with a section titled, Mathematical Practices - A Closer Look that explains a few of the MPs that will be used within that lesson. The MP is identified and an explanation of how to address the MP within the lesson is provided. At times, the identification is targeted, and gives a specific problem where the MP is included, but often it is broad and provides a general statement of how to include the MP within the lesson.

Examples of MPs that are identified and enrich the mathematical content include:

• Lesson 1.2, Ratio Tables and Graphs, “MP1: Students are given the opportunity to make sense of a situation using Exercises 10 through 14. The amount of perseverance needed from students increases throughout this set. Students should ask themselves, “Does my answer make sense?” after reaching an answer for #14.”
• In Lesson 5.4, Evaluating Expressions, “MP2: Have students substitute a variety of values into an expression representing a real-world scenario (like the taxi in Example 3). Have them discuss what the resulting value represents in terms of the situation.”
• In Lesson 3.2, Percents, Decimals and Fractions, “MP5: In Example 3, a ratio table is used as a tool to convert a ratio to a percent. Students have many tools they have learned in previous years that they may choose to connect to new content to help it make sense to them. This should be encouraged.”
• Lesson 7.3, Inequalities, “MP6: When students graph inequalities on a number line, they must pay attention to the inequality sign to determine whether or not the end point is filled in. They must also determine if the arrow should point to numbers greater than or less than the given number. Remind students that each aspect of the graph must be precise to represent the inequality correctly.”

The instructional materials reviewed for EdGems Math Grade 6 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Examples of the student materials prompting students to construct viable arguments and/or analyze the arguments of others include:

• In Lesson 1.4, Comparing Rates, students construct an argument to explain their work. Exercise 16: “Kent and Aleesha each live 5 miles from school and ride their bikes there every day. They left school at the same time on Thursday. After 5 minutes, Aleesha texted Kent that she had traveled 1 mile. After 15 minutes, Kent texted Aleesha saying he was halfway home. If they each traveled at a constant rate, who got home first? Explain how you know.”
• One of the Teacher Gems activities is called “Always, Sometimes, Never.” The instructions for this type of activity state, “Always, Sometimes, Never is best used with concepts that allow for situations that create exceptions to the “rule” or require students to understand subcategories to fully understand the standard. Students are given the opportunity to create evidence to support whether a statement is always true, sometimes true, or never true.” For example, in Lesson 4.4 Multiplying and Dividing Mixed Numbers, the student directions for the Always, Sometimes, Never Activity state, “Decide if the statement in the box below is always true, sometimes true, or never true. Use the remainder of the page to provide mathematical evidence that supports your decision. Statement #1: The quotient of two fractions is the reciprocal of the product of those two fractions.”
• In Lesson 9.2, Area and Perimeter with Decimals, Exercise 22, Students construct an argument to explain their work. “Two rectangles both have perimeters of 20 cm but have different areas. One of the rectangles has dimensions that are not whole numbers. The other rectangle has a length that is 6 cm longer than its width. Draw a set of two possible rectangles that fit these criteria. Show how your set of rectangles meet each of the criteria.”
• In Lesson 1.1, Ratios, Problem 26, students analyze someone’s mathematical reasoning. “Dalexis says, “If you multiply a 2-digit number and a 1-digit number, you get a 3-digit number.” Is her statement always true, sometimes true or never true? Support your answer with your reasoning.”
• In Lesson 2.4, Dividing by Multi-Digit Numbers, Exercise 13, students critique the reasoning of others and justify their thinking. “Vivaan said the answer to #10 above [5240 ÷ 160] should be the same as the answer to 524 ÷ 16. Is he correct? If so, explain why.”
• Unit 3, Percents Performance Task, “Mack wants to sell his used car. He spends the first month trying to sell it for $5,000, but is unsuccessful. The second month he decides to sell the car at 50% off the original selling price. The car does not sell. The third month he discounts the car by an additional 50%. Now, his friend Jasper says he will buy the car because it is free. Jasper’s thinking is below: 50% + 50% = 100% discount = FREE! 1. Explain what is wrong with Jasper’s thinking. Include the percent of Mack’s original selling price Jasper would pay for the car if he buys it. Use words and/or numbers to show how you determined your answer.” Students critique the reasoning of others and justify their reasoning. 
• Lesson 3.4, Percent Applications, students analyze a student’s work and provide mathematical reasoning to correct the strategy in Problem 14 of the student lesson, “Mikayla took her friends to lunch. The bill came to $32. She wanted to leave a 15% tip and needed to determine the total cost of the lunch, including the tip. Her work is below. Unfortunately, Mikayla made a mistake. Explain the mistake Mikayla made and then find the correct total for the bill.”

The instructional materials reviewed for EdGems Math 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. The Teacher/Lesson Guide and Teacher Gems within most lessons support teachers to engage students in constructing viable arguments and analyzing the reasoning of others. Examples include:

• Lesson 1.1, Ratios, Teacher/Lesson Guide, Exercise 19b, supports teachers in engaging students in constructing viable arguments. “Exercise 19b could be used as an opener to the lesson as it builds upon previous knowledge. Groups can discuss if the situation is possible and create arguments for or against to share with classmates.” 
• In Lesson 3.2, Percents, Decimals and Fractions, Teacher/Lesson Guide, Explore!, teachers engage students to construct an argument and justify their thinking. “In Step 5 of the Explore!, students will create differently shaded grids. Have them share their work with one another or as a class using a document reader and discuss whether or not the shaded grids accurately reflect the furniture in Tamika’s room. Have students emphasize their work in Step 4 as justification for their grids.”
• Lesson 5.1, Powers and Exponents, Teacher/Lesson Guide, supports teachers to help students construct viable arguments. “Show one correct and one incorrect statement (e.g., 3⁴ = 81 and 3⁴ = 12). Ask students to find the correct statement and construct a viable argument as to why it is correct.”
• Lesson 5.4, Evaluating Expressions, Teacher/Lesson Guide, supports teachers to help students critique the reasoning of others. Teachers are informed of a misconception: “It is very common for students to insert values into expressions using multiplication and lose the operation (i.e., 5y when y = 2 becomes 52).” Teachers are instructed to, “Show this common error on the board and see if anyone can figure out what you did wrong.”
• In Lesson 6.1, Equations and Solutions, Teachers Gems, Partner, teachers assist students in constructing viable arguments and critiquing the reasoning of others. “After a rotation, if there is a lot of discrepancy in student answers as they compare with their new partner, the teacher can call a FREEZE. With a FREEZE, all students should put their writing utensils down then ask for a partner set where they got different answers on a specific task. These students hand over their templates for the class to examine (under a document camera) and provide feedback on. Having classroom sentence starters for the FREEZE component can be helpful in guiding the conversation such as “I like… I wonder… Your next step could be…”

The instructional materials reviewed for EdGems Math Grade 6 meet expectations for explicitly attending to the specialized language of mathematics. The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Throughout the materials, precise terminology is used to describe mathematical concepts, and each lesson includes a visual lesson presentation. In most of the lesson presentations, there is at least one slide dedicated to explicit teaching of vocabulary. The Teacher/Lesson Guide, Student Lesson, and Parent Guide all contain information about mathematical language. Examples include:

• In the Student Lessons, the font for vocabulary words is red and a definition is included.
• Each unit includes a Parent Guide which contains “Important Vocabulary” related to the unit.
• In Lesson 1.2, Ratio Tables and Graphs, the Lesson Presentation includes a slide dedicated to introducing vocabulary from the lesson and provides mathematical definitions. “Equivalent Ratios - A comparison of two quantities with the same value. Ratio Table - A model that shows multiple equivalent ratios."
• In Lesson 6.1, Equations and Solutions, the Lesson Presentation includes a slide that provides the definitions and terms important to the lesson. “Equation - A mathematical sentence that contains an equals sign (=) between two equivalent expressions. Solution - Any value or values that makes an equation true.”
• Lesson 5.1, Powers and Exponents, Teacher/Lesson Guide, Teaching Tips includes information about mathematical language, “People often use the word “squared” for power of 2 and “cubed” for power of 3. For example, students should understand that four squared means four to the power of 2, or 4².”
• Lesson 7.1, Understanding Integers, Teacher/Lesson Guide, Teaching Tips includes information about mathematical language. “Addition and subtraction are inverse operations that “undo” one another. Positive and negative integers are opposites of one another. Be careful not to say addition and subtraction are opposites as this may confuse students.”
• In Lesson 2.3, Dividing by 1-Digit Numbers, Online Practice, students identify a number based on accurate terminology. “In each expression determine if 4 is the ‘dividend’ or the ‘divisor’.”

Examples of not attending to the specialized language of mathematics include:

• In Lesson 6.3, Solving Multiplication and Division Equations, Teacher/Lesson Guide, “MP6: In a division equation, students may struggle with seeing how multiplying both sides by the number in the denominator cancels out the value in the denominator. Show students the importance of writing the value they are multiplying by in the numerator, not in the denominator.” “Cancels out the value in the denominator” is not precise mathematical terminology.
• In Lesson 2.1, Adding and Subtracting Decimals, Teacher/Lesson Guide, Teaching Tips, “Remind students, when adding or subtracting decimals, that once the problem is lined up properly the decimal points are ignored until the end when it is brought straight down and placed in the answer.” This teaching tip is not mathematically sound. Students should be conceptually aware of why a process is occuring versus simply stating a procedure. 
• In Lesson 5.5, Equivalent Expressions, Teacher/Lesson Guide, “MP6: When combining like terms, help students attend to precision by understanding that the operation preceding each term ‘belongs’ to that term. When a term is being subtracted from a previous term, the subtraction sign should move with that term as a negative sign.” Student Lesson: “In order to simplify an algebraic expression you must combine all like terms. When combining like terms you must remember that the operation in front of the term (addition or subtraction) must stay with the term. Rewrite the expression by grouping like terms together before adding or subtracting the coefficients to simplify.” In both the Lesson Guide and Student Lesson, the explanation for combining like terms and the idea that the operation “belongs” to the term is not mathematically sound. Students should be conceptually aware of why a process is occurring versus simply stating a procedure.