6th Grade - Gateway 2

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards of cluster headings.

Materials develop conceptual understanding throughout the grade level, providing opportunities for students to independently demonstrate their understanding through various program components. Each lesson incorporates an Explore! activity for students to discover new concepts using diverse methods, a Lesson Presentation with slides that supports reasoning and sense-making through examples and communication breaks, a Student Lesson featuring mathematical representations, and a selection of Teacher Gems designed to target conceptual understanding through engaging activities such as Always, Sometimes, Never, Categories, Four Corners, and Climb the Ladder. Examples include: 

• Unit 2, Lesson 2.1, Explore!, students use tape diagrams to develop a conceptual understanding of ratios (6.RP.1). In Step 1, four ratios are provided: "Granny Apple Green: 2 parts blue to 3 parts yellow, Orange Delight: 1 part red to 2 parts yellow, Peaceful Purple: 5 parts blue to 4 parts red, and Totally Teal: 4 parts blue to 1 part green." Step 2, “A tape diagram is shown below for Granny Apple Green paint. Draw a tape diagram for each of the other colors.”

• Unit 5, Lesson 5.5, Explore!, students use the properties of operations and substitution to develop conceptual understanding of equivalent expressions (6.EE.3). An example provided is as follows: “Step 1: Two expressions are equivalent if they have the same value. Connect each expression on Line A with its equivalent expression on Line B. Line A: 7^{2}-1, 4(6-1), 4+2(9). Line B: (2+1)^{2}+11, 10(5)-2, 6^{2}\div 2+4. Step 2: Two of the four expressions listed below are equivalent. Which expressions do you think are equivalent? Explain your reasoning. 2x + 2, 3(x + 1) - 2, 3x + 1, 2x + x + 2. Step 3: Algebraic expressions are equivalent expressions if any value for the variable is substituted into both expressions and the expressions simplify to the same value. Choose a value for x and substitute it into both expressions you chose in Step 2. Do they equal the same amount when you evaluate the expressions? If not, pick a different pair of expressions from Step 2 and test these expressions.”

• Unit 10, Lesson 10.6, Lesson Presentation, Slide 4, Explore!, students calculate the mean and median, in reference to a context, to develop conceptual understanding of using the measure of center that best represents a data set (6.SP.5.C). An example provided is as follows: “Mr. Hinton decided to analyze the number of problems he gave for homework during the last unit. He gave a total of nine assignments. The assignments had the following number of problems: 10, 8, 12, 18, 9, 10, 17, 24, 8. Step 1: What is the mean number of problems students were given per homework assignment? Step 2: What is the median number of problems on the nine assignments? Step 3: Which statistic, mean or median, do you feel better represents the data set?”

The materials provide students with opportunities to engage independently with concrete and semi-concrete representations while developing conceptual understanding. Examples include:

• Unit 3, Lesson 3.1, Leveled Practice-P, Exercises 1-3, students demonstrate conceptual understanding of percent as they use models to represent ratios out of 100 as a percent (6.RP.3.C). An example provided is as follows: “For each shaded grid, write the ratio of the shaded squares to 100 (as a fraction) and the percent of squares shaded as a number with the % sign.” Three grids of 100 squares are given. The first set has 30 shaded squares, the second set has 24 shaded squares, and the third set has 25 shaded squares.

• Unit 6, Lesson 6.2, Student Lesson, Exercise 10, students demonstrate conceptual understanding of equations as they represent situations with an equation and explain the meaning of the solution (6.EE.7). The example is as follows: “Carrie spent $32 at the movie theater. She had $15 left in her wallet. a. If y represents the original amount of money Carrie took to the movie theater, explain why the equation y − 32 = 15 represents this situation. b. Solve this equation using inverse operations. c. What does the answer to this equation represent?”

• Unit 7, Lesson 7.1, Student Lesson, Exercise 18, students demonstrate a conceptual understanding of integers as they create a context that requires a negative integer (6.NS.5). An example provided is as follows: “Create a situation that would be represented by a negative integer. Explain why a negative integer makes sense for your situation.”

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

There are opportunities for students to develop procedural skills and fluency in each lesson. The materials support the development of these skills and fluencies through Starter Choice Boards, Student Gems, Lesson Examples, Student Exercises, and Teacher Gems. Examples include:

• Unit 1, Lesson 1.2, Teacher Gem: Always, Sometimes, Never, develop procedural skill and fluency for multiplying multi-digit decimals as they give examples as evidence to support their decisions regarding statements of place value (6.NS.3). An example is as follows: “Decide if the statement in the box is always true, sometimes true, or never true. Use the remainder of the page to provide mathematical evidence that supports your decision.” Statements include: “Statement #1: When multiplying decimals, the amount of digits before a decimal point in a number affects the placement of the decimal point in the answer. Statement #2: The product of a whole number and a decimal is smaller than the whole number. Statement #3: Multiplying by a decimal is the same as multiplying by a fraction equivalent to that decimal. Statement #4: The product of two numbers, each with two digits to the right of a decimal point, will have more than two decimal places in the answer. Statement #5: 1. \square \times 0.\square is greater than 1.”

• Unit 2, Lesson 2.4, Starter Choice Board, StoryBoard Starter, Building Blocks, Exercises 1-4, students develop fluency in equivalent ratios as they compare ratios as fractions using <, >, = (6.RP.1). “Use <, >, or = to compare the following fractions. 1. \frac{3}{5}\square\frac{9}{15} . 2. \frac{5}{7}\square\frac{5}{9}. 3. \frac{1}{2}\square\frac{2}{3}. 4. \frac{7}{10}\square\frac{3}{4}.”

• Unit 5, Planning and Assessment, Launch and Finale Teacher Gems, Pathways Finale, students develop procedural skill and fluency in writing and evaluating numerical expressions with whole-number exponents as they write and evaluate expressions (6.EE.1). The activity can be done in a small group center with the teacher or with expert students the teacher has identified. An example provided is as follows: “Skill 1: I can write and compute expressions with powers and evaluate expressions using the order of operations. Skill 1A: Find the value of 3^{4}. Skill 1B: Write 5\times 5\times 5\times 5\times 5\times 5 as a power. Skill 1C: Find the value of the expression. 2^{3}\times(5+2)+4. Skill 1D: Find the value of the expression. \frac{(4\times 2)^{2}}{11-7}.”

There are opportunities for students to develop procedural skill and fluency independently throughout the grade level. Examples include:

• Unit 3, Lesson 3.2, Exit Card, students independently demonstrate procedural skill and fluency as they convert fractions and decimals to percents (6.RP.3c). An example provided is as follows: “1. Write each decimal as a percent. a. 0.32 b. 0.1c. 3.5 2. Write each fraction as a percent. a. \frac{3}{50} b. \frac{1}{4} c. 4\frac{1}{3}” 

• Unit 6, Planning and Assessment, Readiness 1, Exercises 1-3, students independently demonstrate procedural skill and fluency as they determine an unknown in an equation (6.EE.5). An example provided is as follows: “Determine the unknown number that makes the equations true in each of the equations. 1. 9 × ⬛ = 54. 2. 6 = ⬛ ÷ 5. 3. ⬛ + 7 = 18.” 

• Unit 9, Lesson 9.4, Student Lesson, Exercise 2, students independently demonstrate fluency as they multiply decimals to find the volume of a rectangular prism with decimal lengths (6.NS.3). An example provided is as follows: “Find the volume of each rectangular prism.” A drawing of a cube with side lengths of 4.6 cm is provided.

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

There are opportunities for students to develop routine and non-routine applications of mathematics in each lesson. The materials develop application through the Student Lesson Exercises in the Apply to the World Around Me section, Teacher Gems, a Storyboard Launch/Finale, and Performance Tasks.

Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 2, Lesson 2.2, Student Lesson, Example 1, students use ratio and rate reasoning with tables and graphs to solve a real-world ratio problem (6.R.3). Example 1 states, “Petra saves $3 for every $1 she spends. a. Complete the ratio table to show the relationship between Petra’s amount saved compared to her amount spent.” A ratio table with values of 3 and 15 for “Amount Saved” and 2 and 3 for “Amount Spent” is given, with corresponding values blank. b. Create a graph to model this relationship. c. What does the point (15, 5) represent in this situation?”

• Unit 6, Lesson 6.1, Student Lesson, Exercise 5, students write and solve non-routine mathematical problems using equations in the form of x+p=q and px=q (6.EE.7). An example states, “Write four equations involving variables, each using a different operation (add, subtract, multiply, and divide), that are true when 5 is substituted for the variable. Use mathematics to justify your answer.”

• Unit 9, Lesson 9.3, Student Lesson, Exercise 7, students solve a non-routine mathematical problem involving surface area (6.G.4). An example states, “Every dimension of a rectangular prism is doubled. How does the new surface area compare to the original surface area? Explain your reasoning.”

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

• Unit 4, Planning & Assessment, Performance Assessment, Exercise 2, students independently solve non-routine word problems involving division of fractions (6.NS.1). Exercise 2 states, “Pete sold 8 out of 16 candy bars on the first day of a fundraiser. He sold \frac{3}{4} of the candy bars had left on the second day. On the third day he sold \frac{1}{2} of the candy bars he had left. His sister congratulated him on selling all his candy bars. Did Pete actually sell all his candy bars? Show all work necessary to justify your answer.”

• Unit 4, Lesson 4.4, Student Lesson, Exercise 11, students independently solve routine problems involving the division of fractions by fractions (6.NS.1). Exercise 11 states, “Ty edged a flower bed with paving stones. Each paving stone was 5\frac{3}{4} inches long. The length of the flower bed was 4014 inches long. How many paving stones did Ty need?“ 

• Unit 10, Lesson 10.6, Leveled Practice T, Exercise 1, students independently solve a routine word problem and find and use measures of center and variability for the two data sets (6.SP.2). Exercise 1 states, “Brittany was shopping for CDs at two different stores. The prices for various CDs at the two stores are shown below. Store A: $9, $12, $14, $20, $20 Store B: $12.50, $13, $16, $16, $17.50 a. Find the mean CD price for each store. Mean (Store A = ____) Mean (Store B) = _____ b. How do the means compare: c. Find the range of the CD prices for each store. Range (Store A = ____) Range (Store B) = _____ d. What do the ranges tell you about the price of CDs at each store? e. Why is only comparing the means of the data sets misleading? f. Which store would you prefer to buy your CDs at? Explain.”

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade level. 

All three aspects of rigor are present independently throughout each grade level. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 1, Lesson 1.1, Leveled Practice P, Exercise 7, students demonstrate fluency as they apply multi-digit decimal operations to real-world problems (6.NS.3). Exercise 7 states, “Celia bought a camera for $158.96 and a scrapbook for $38.75. How much did she spend altogether?”

• Unit 4, Lesson 4.4, Student Lesson, Exercise 18, students apply their understanding by computing quotients of fractions in a word problem (6.NS.1). Exercise 18 states, “Clint needs to make a platform that is \frac{3}{4} inch thick. He has boards that are each \frac{3}{8} inch thick. How many boards does he need to make the platform?”

• Unit 6, Lesson 6.4, Leveled Practice P, Exercise 11, students deepen their conceptual understanding of mathematical problems by applying their knowledge of percentages and proportional relationships to determine the range of original prices based on the given discount amounts (6.RP.3c). Exercise 11 states, “All clearance swimsuits were marked down 60%. One rack of swimsuits had a sign stating, ‘Save $18 to $27 on these suits!’ What was the range of the original prices of the swimsuits on this rack?”

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

• Unit 1, Lesson 1.5, Student Lesson, Exercise 8, students use all three aspects of rigor, conceptual understanding, procedural fluency, and application as they analyze and solve decimal division problems, evaluate the effectiveness of strategies, and apply their knowledge to new situations (6.NS.3). Exercise 8 states, “Paola is trying to find the value of 73.8 divided by 8.2. Alvaro says, ‘You can just remove the decimals from both the dividend and divisor and divide as if they are whole numbers to get the answer.’ a. Will Alvaro's strategy work for this problem? Why or why not? b. Give an example of a decimal division problem in which Alvaro's strategy would not work. Explain why.” 

• Unit 6, Lesson 6.4, Leveled Practice T, use all three aspects of rigor, conceptual understanding, procedural fluency, and application as they find percents of quantities as rates per 100 and solve real-world problems involving the whole, given a part and the percent (6.RP.3c). Practice T states, “1. Dina saved 30% of all the money she made over the summer weeding at her grandma’s house. She saved $60. a. If $60 is 30%, what value is 10%? Fill the value in on the double number line above 10%. b. Use the value above 10% to find the total value (above 100%) by multiplying by 10. Fill it in on the double number line. How much money had Dina earned weeding in all?” A double number line is provided with “Dollars” labeled on the top line and “Percent” labeled on the bottom line. The percents are marked in 10% increments, with $60 labeled above 30%. “Solve each percent problem for the unknown. 2. 50% of ___ is 8 4. 20 is ____% of 200. 7. 0.7 is 5% of _____.”

• Unit 8, Lesson 8.2, Student Lesson, Exercise 10, students use all three aspects of rigor, conceptual understanding, procedural fluency, and application as they graph linear equations, create input-output tables, and justify their reasoning to determine whether the graph forms a straight line. (6.EE.9). Exercise 10 states, “In this lesson, all the equations you examined and created are linear equations, which means their graphs form lines. Graph the equation y=2^{x} using an input-output table with x-values 0, 1, 2, 3, 4 and 5. Is this graph a linear equation? Explain your reasoning.”

The materials reviewed for EdGems Math (2024) Grade 6 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. Students have opportunities to engage with the Math Practices throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

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

• Unit 1, Lesson 1.6, Student Lesson, Exercise 13, students make sense of problems by determining the least common multiple to find the equal quantities needed and use this information to calculate the total price. Exercise 13 states, “A caterer buys cinnamon rolls in boxes of 6 and bagels in boxes of 9. Each cinnamon roll box costs her $3.75 and the bagel boxes cost $2.25 each. She wants to purchase the least number of boxes possible to have an equal amount of cinnamon rolls and bagels. What will her total cost be?”

• Unit 3, Lesson 3.4, Lesson Presentation, Example 2, with the support of the teacher, students make sense of real-world percent situations by interpreting and solving problems using percent calculations, persevering through multi-step processes. Example 2 states, “Graham traveled to Florida to visit family. While in Florida he bought a shirt priced at $25.00. The sales tax in Florida was 6%. What was the total amount he paid for the shirt? Write the problem. 6% of $25.00 is what? Solve the percent problem. 0.06\times25.00=1.5 Sales tax on $25.00 is $1.50. Add the tax to the original price. 25.00 + 1.50 = 26.50. The total amount for the shirt, including tax, was $26.50.”

• Unit 7, Lesson 7.2, Student Lesson, Exercise 9, students make sense of problems by comparing and ordering rational numbers, using strategies such as number line placement, equivalent forms, and reasoning about magnitude. Exercise 9 states, “Give a fraction that is greater than -\frac{5}{6} but less than -\frac{2}{3}. Explain how you know your fraction fits these criteria.” 

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

• Unit 4, Lesson 4.3, Student Lesson, Exercise 9, students reason abstractly and quantitatively as they interpret and explain the value of the quotient when dividing fractions, and provide an example to support their reasoning. Exercise 9 states, “How does the quotient compare to the dividend when the divisor is a fraction between 0 and 1? Give an example to support your answer.” 

• Unit 5, Lesson 5.4, Student Lesson, Exercise 8, students reason abstractly and quantitatively as they create and evaluate expressions with given conditions. Exercise 8 states, “Create three different expressions using x, y and/or z that have a value of 100 if x = 5, y = 2 and z = 25. Use mathematics to prove your expressions have a value of 100.” 

• Unit 7, Lesson 7.3, Explore! students reason abstractly and quantitatively as they identify values to make true statements, write inequalities to represent situations, and describe a graph of an inequality. The materials state, “Step 1: Read each statement and circle all the possible values that make the statement true. a. The low today was less than 0! -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. Step 2: Write two more numbers that are not integers that would make the statements true. Step 3: Choose one of the statements above. How many possible values could make the statement true? Explain your reasoning. Step 4: You can use a variable and an inequality symbol to represent each of the statements from Steps 1-2. Make an attempt to write an inequality for each statement. Use x as the unknown value. Part a has been done for you. b. “She has at least $3 in her pocket.” Step 5: A graph of an inequality shows all the possible values that make the statement true. What do you think the graph of x < 0 might look like?”

The materials reviewed for EdGems Math Grade 6 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with the Math Practice throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

Students construct viable arguments, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 3, Materials, Performance Task, students construct a viable argument when they justify their answer to a percent discount problem. The materials state, “Part 3: Donnell found the original prices of the shoes he likes. 4. Which pair of shoes should Donnell get? Justify your answer with words, symbols and mathematics.“ A table is provided, showing four pairs of shoes with the following original prices: $345, $185, $245, $170.

• Unit 5, Lesson 5.1, Student Lesson, Exercise 23, students construct a viable argument as they justify the steps and reasoning in a mathematical process, using evidence from calculations or known properties to support their conclusion. Exercise 23 states, “Aisha and Julio were shopping together. Aisha wanted to buy an item that was 40% off of $50. She said, “I can find out how much I will have to pay by finding 60% of $50.” Explain why Aisha’s method would work.”

• Unit 8, Lesson 8.1, Student Lesson, Exercise 8, students construct a viable argument as they clearly articulate their mathematical reasoning, connect it to relevant concepts or properties, and use evidence from their calculations or examples to justify their conclusions. Exercise 8 states, “Rosa claims if one of the variables in a relation is a unit of time, then time is the independent variable. Is Rosa’s claim always true, sometimes true or never true?”

Students critique the reasoning of others, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Lesson 2.2, Student Lesson, Exercise 8, students critique the reasoning of others as they explain the relationship between ratios and ordered pairs on a coordinate plane. Tanner stated, Exercise 8 states, “When equivalent ratios are graphed as ordered pairs on a coordinate plane, the points fall into a straight line.” Is his statement always true, sometimes true or never true? Explain your reasoning.”

• Unit 7, Lesson 7.3, Student Lesson, Exercise 13, students critique the reasoning of others by carefully analyzing the mathematical arguments presented, identifying any flawed assumptions or steps, and offering corrective feedback or alternative approaches based on valid mathematical principles. Exercise 13 states, “Wilson told Joey, ‘There are an infinite number of positive values for x that make the statement x< 10 true.’ Joey disagreed. Joey said the only values that make that statement true are 1, 2, 3, 4, 5, 6, 7, 8 and 9. Who do you agree with? Why?”

• Unit 9, Lesson 9.1, Student Lesson, Exercise 7, students critique the reasoning of others as they compute with decimals to solve for perimeter. Exercise 7 states, “A rectangle is 12.4 cm by 8.2 cm. Mailan says if she cuts a piece off the rectangle it will have a perimeter less than 41.2 cm. Is her statement always, sometimes or never true? Explain your reasoning.”

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for supporting the intentional development of MP4: Model with mathematics and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to solve real-world problems, model situations with appropriate representations, and describe what they do with the model and how it relates to the problem. Students model with mathematics as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 3, Lesson 3.3, Student Lesson, Exercises 10, students model with mathematics when they use a double number line to solve a percent discount problem. Exercise 10 states, “A bicycle was $120. It was on sale for 30% off. Use your completed double number line from Exercise 1 to determine the new cost of the bicycle.“

• Unit 4, Materials, Performance Task students model with mathematics as they use division of fractions in a real-world problem. The materials state, “Part 1: Each week, Jerome bakes and sells peanut butter chocolate chip cookies to his neighbors. He is saving up to purchase a new set of wireless headphones for $199. Jerome’s orders for this week are shown below:” A table is given with family names and the amount of cookies they order in dozens: 1\frac{1}{2}, 4\frac{1}{4}, 2\frac{1}{2}, 5\frac{5}{12}, 3\frac{1}{6}, 2, 5\frac{1}{2} “1. What are some questions related to the situation above that could be solved using mathematics? 2. What additional information would you need to solve the questions you created? Part 2: Jerome charges $4 per dozen cookies. 3. Will Jerome be able to purchase the headphones using only his earnings from this week? Justify your answer using words and mathematics.” 4 per dozen cookies. 3. Will Jerome be able to purchase the headphones using only his earnings from this week? Justify your answer using words and mathematics.” 

• Unit 8, Lesson 8.1, Lesson Presentation, Example 2, with the support of the teacher, students represent everyday situations using models and other representations to create an input-output table for equations with two variables. Example 2 states, “Payton received a small kitten for her birthday. The kitten weighed 1 pound. Each week, the kitten gained 0.5 pounds. The weight of the kitten, y, can be found using the equation y=1+0.5x where x is the number of weeks she has had the kitten. a. Complete the table for the first 5 weeks Payton had the kitten. b. Write five ordered pairs by pairing the input and output values. c. Graph the ordered pairs on a coordinate plane.” 

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to identify and use a variety of tools or strategies, working with the support of the teacher and independently, throughout the units to support their understanding of grade-level math. Examples include:

• Unit 2, Materials, Performance Task, Part 2, students use appropriate tools as they show at least two different strategies of their choosing for solving a real-world ratio problem. Performance Task , Part 2 states, “Santiago measured the aquariums and labeled the dimensions (as seen below). He filled Aquarium A with water in 20 minutes.” Students are given an image of two aquariums, Aquarium A with dimensions 18in\times3ft\times12in and Aquarium B with dimensions 2ft\times5ft\times18in. “5. If Santiago’s co-worker is dropping off the fish in one hour, will Santiago have Aquarium B filled in time? Use at least two different strategies to support your answer.” 

• Unit 3, Lesson 3.4, Teacher Gems: Masterpieces, students use appropriate tools as they choose a strategy to solve real-world percent problems. The materials state, “Kyle designed a spinner. Each section of the spinner is equal in size. Kyle makes 30% of the sections blue, 25% of the sections red and 20% of the sections yellow. The remaining sections of the spinner are green. 1. Five sections of the spinner are red. How many total sections are on the spinner? 4. Lacey wants to create a spinner that has the same four colors as Kyle’s spinner but meets the following criteria: Has 15 total equal-sized sections. The yellow and blue sections combined outnumber the total number of red and green sections. There are three times as many blue sections than green sections. The blue sections make up 40% of the spinner. There is one less green section than red. Describe a spinner that meets Lacey’s criteria and EXPLAIN/SHOW how it meets each of the criteria.”

• Unit 9, Lesson 9.2, Lesson Presentation, Explore!, with the support of the teacher, students use appropriate tools as they determine a strategy to solve a composite area problem. The materials state, “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. Step 2, Is there only one way to find the area of the polygon above? Explain your reasoning. Step 3, Kienan’s boss told him to quote for 1 cubic yard of bark per 108 square feet. He also told him to round up to the next whole cubic yard of bark. How many cubic yards of bark will the park need? Step 4, Kienan needs to prepare a quote for the owners of the park. Each cubic yard of bark costs $38. There is also a $25 delivery fee. How much should he quote for the bark and delivery?” 

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have many opportunities to attend to precision and the specialized language of math, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Lesson 2.3, Student Lesson, page 119, students attend to the specialized language of mathematics when defining rate and unit rate in their own words. The Student Lesson states, “A rate is a comparison of two numbers with different units. For example, Deidre’s Donut Shop charges $6.00 for 12 donuts. A unit rate is a rate that can be written as a fraction with a denominator of 1. Unit rates can also be written as a single number using the work ‘per’. Unit rates can be found by creating an equivalent fraction with a denominator of 1.” All three expressions are identified as rates, and the second and third expressions are identified as unit rates. “In my own words… A rate is … In my own words… A unit rate is …”

• Unit 3, Lesson 4.3, Student Lesson, Exercises 9, students attend to the specialized language of mathematics as they compare the quotient and dividend. Exercise 9 states, “How does the quotient compare to the dividend when the divisor is a fraction between 0 and 1? Give an example to support your answer.”

• Unit 6, Lesson 6.2, Student Lesson, Exercise 3, students attend to precision when solving equations and checking their answers. Exercise 3 states, “Solve each equation. Check your solution. p+\frac{1}{4}=2\frac{1}{2}.” 

• Unit 9, Lesson 9.3, Student Lesson, Exercise 10, students attend to precision as they calculate surface area and compare production costs for two rectangular prisms. Exercise 10 states, “Material used to make plastic storage boxes cost $0.02 per square inch. Box A has dimensions of 12 inches by 8 inches by 6 inches. Box B is a cube with dimensions of 10 inches. How much more will it cost to make Box B compared to Box A?”

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 3, Lesson 3.1, Lesson Presentation, Explore!, students, with the support of the teacher, make use of structure as they demonstrate understanding of percentages as a rate per 100. The materials state, “Step 1 For each shaded grid, write: The ratio of the shaded squares to 100 (as a fraction). The percent of squares shaded as a number with the % sign.” Three examples are provided and shaded for students to determine the ratio and percent. “Step 2 How many squares would be shaded on a 10 by 10 grid for each percent given below? a. 1% b. 25% c. 50% d. 100% e. 0%. Step 3 Kim bought 100 balloons for her birthday party. She used 86 of them. What percent of the balloons did she use? Step 4 C.J. used 60 envelopes out of 100. a. What percent of envelopes did he use? b. What percent of envelopes were left over? Step 5 Shade in \frac{3}{10} of the squares in the grid at the right. Step 6 What percent of the squares are shared in the grid from Step 5? Step 7 The percent from Step 6 and the fraction \frac{3}{10} are equivalent. Explain how you could determine this without shading in a grid.” 

• Unit 4, Lesson 4.4, Teacher Gems, Four Corners, students look for structure as they create representations when given an expression, application situation, picture model or a quotient involving division of fractions. The materials state, “Jill has 3\frac{2}{3} cups of flour. She has a measuring cup that holds \frac{1}{3} cup. How many scoops will it take to use all the flour?” Students must create a related expression, model, and quotient. 

• Unit 6, Lesson 6.2, Student Lesson, Exercise 12, students look for structure as they identify errors in solved equations. Exercise 12 states, “Jordan solved the two equations below incorrectly. Explain what she did wrong for each equation.” Two solved equations are shown. For Equation A, x + 129 = 356, 129 is subtracted from one side and added to the other, resulting in x = 485. For Equation B, 4.98 = x − 2.7, 2.7 is subtracted from both sides, resulting in x = 2.28.

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to use repeated reasoning in order to make generalizations and build a deeper understanding of grade-level math concepts as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 1.4, Student Lesson, Exercise 10, students look for and use repeated reasoning to calculate volume and divide it equally among smaller prisms. Exercise 10 states, “A rectangular prism was 14 inches by 9 inches by 22 inches. The prism was filled with sand and then the sand was used to fill up 11 equal-sized smaller prisms. a. How many cubic inches of sand are in each of the smaller prisms? b. Give possible dimensions of the smaller prisms.”

• Unit 2, Lesson 2.2, Lesson Presentation, Example 1, students, with the support of the teacher, look for and use repeated reasoning as they create a ratio table showing equivalence using repeated addition or multiplication. Example 1 states, “Petra saves $3 for every $1 she spends. a. Complete the ratio table to show the relationship between Petra’s amount saved compared to her amount spent. Using repeated addition, add 3 to each preceding value in the top row and add 1 to each value in the bottom row.” A table is shown with the label 'Amount Saved' and values 3, 6, 9, 12, and 15, and 'Amount Spent' with values 1, 2, 3, 4, and 5. The 'Amount Saved' row shows jumps of +3 between each value, while the 'Amount Spent' row shows jumps of +1 between each value. Example 2 states, “The Hines Parks Service plants 16 birch trees for every 40 oak trees. Use the ratio table to determine how many birch trees will be planted when 55 oak trees are used. Create a simplified equivalent ratio using division. Use multiplication to find the missing value.” A table is shown with the label 'Birch Trees' and values 16, 2, and 22, and 'Oak Trees' with values 40, 5, and 55. The Birch Trees row shows jumps of 8 and 11 between the values, and the Oak Trees row also shows jumps of 8 and 11 between the values.

• Unit 10, Lesson 10.6, Student Lesson, Exercise 7, students look for and use repeated reasoning as they display data in a dot plot and give quantitative measures of center. Exercise 7 states, “Mr. Tobin and Mrs. Vicente compared their students’ scores on their latest quiz. The results are shown below. Mr. Tobin: 8, 6, 9, 1, 3, 10, 5, 1, 7, 9, 2, 10. Mrs. Vicente: 3, 7, 9, 2, 7, 6, 1, 4, 10, 7, 8, 4 a. Make a dot plot for each teacher’s scores. b. Describe the differences in the two dot plots. c. Find the measures of center for each teacher. How do they compare? d. Find the five-number summary for each teacher. e. How does the IQR for Mr. Tobin’s class compare to the IQR for Mrs. Vicente’s class? What does this tell you about the spread of their data?”