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?”

The materials reviewed for EdGems Math (2024) Grade 6 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 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:

• Key instructional support through resources designed to enhance teacher effectiveness. The Unit Planning & Assessment pages offer access to both general course and unit-specific instructional information, ensuring teachers have the necessary materials for lesson execution. The PD Library includes written and video-based professional development on implementing Teacher Gems, Communication Breaks, Fluency Boards & Routines, and the 5E Instructional Model, equipping teachers with techniques for effective instruction. Additionally, the ELL Supports Guide provides strategies for ELL Proficiency Levels, Instructional Design, Mathematical Language Routines (MLRs), and Scaffolding Techniques. This guide includes resources such as a Word Problem Graphic Organizer, Target Trackers, Math Practice Trackers, a Math Self-Assessment Rubric, and a Vocabulary Journal Format, ensuring multilingual learners receive appropriate language supports.

• Lesson planning guidance is structured through unit resources that outline daily instructional expectations. The Unit Launch Guide provides a two-day lesson plan for introducing each unit, detailing required and optional components with class time allocations and facilitation instructions. These components include the Target Tracker Launch, Storyboard Launch, Fluency Board Launch, Readiness Check, and Unit Launch Teacher Gem, all designed to establish foundational knowledge. The Unit Finale Guide supports teachers in unit review, differentiation, and assessment through a three-day lesson plan incorporating the Unit Review, Unit Finale Teacher Gem, Fluency Board Finale, Storyboard Finale, and Assessments, along with explanations of assessment options.

• Lesson implementation support is embedded within the Teacher Guides, which contain detailed two-day lesson plans with structured guidance on instruction and differentiation. The At a Glance section provides a one-page lesson summary covering Standards, Materials, Starter Choice Board, Lesson Planning Overview, and Learning Outcomes. The Deep Dive section offers explicit lesson planning guidance, outlining both required and optional components with recommended class time. Day 1 lessons include the Starter Choice Board, Explore! Activity, Lesson Presentation, and Independent Practice, while Day 2 includes the Starter Choice Board, Teacher Gem options, Exit Card & Target Tracker, and additional Independent Practice. The Deep Dive also incorporates formative assessments, Focus Math Practices, Math Practices: Teacher and Student Moves, and Supports for Students with Learning and Language Differences, ensuring teachers have clear implementation strategies for diverse learners.

Materials include sufficient and useful annotations and suggestions that are embedded within specific learning objectives to support effective lesson implementation. Preparation materials, lesson narratives, and instructional supports provide teachers with structured lesson planning guidance, differentiation strategies, formative assessment recommendations, and opportunities for student engagement. These supports are found in resources such as the Unit Launch Guides, Unit Finale Guides, Lesson Planning Guidance, Teacher Guides, Deep Dive sections, Starter Choice Boards, and Small Group Instruction recommendations.

• Unit 2, Planning & Assessment, Unit Launch Guide, Lesson Planning Guidance: Day 1, “Target Tracker Launch (10 minutes) Have students open their Interactive Textbook to the Target Tracker at the beginning of the unit or, if using notebooks/binders, consider printing the Target Tracker and having students use it as the starting page of their notes for the unit. Ask students to read through the five targets in the unit and UNDERLINE math words they recognize and CIRCLE math words that are new to them. Have students share with a partner a word that they underlined and what they think it means. Consider calling on students to share what their partner said. Create an initial vocabulary recognition list on the board as students share out. Lesson Targets Lesson 2.1: I can simplify and write ratios three ways. Lesson 2.2: I can create ratios that represent the same value. Lesson 2.3: I can use equivalent rates to solve problems. Lesson 2.4: I can compare rates to solve problems. Lesson 2.5: I can convert measurements within systems and between systems.”

• Unit 3, Planning & Assessment, Unit Finale Guide, Lesson Planning Guidance: Day 2, “Storyboard Finale (15 minutes) The Storyboard Finale is the culminating task for the storyline about growing a garden. Students will use percents to calculate markups and discounts. Read the Storyboard together and then give students 5-7 minutes to work independently. Encourage students to use this time to write down key information and possibly a sentence frame at the bottom of the page for their answer before strategizing how to solve. After 5-7 minutes, have students join with a partner to share their thinking and continue solving. Finish by having two partner sets join together to compare and contrast processes and solutions. NOTE: Using Unit Review data, consider calling some students who need additional support into small group instruction during the Storyboard Finale time. Utilize the Pathways cards from the previous day or repurpose Exit Cards from each lesson to review necessary targets.”

• Unit 4, Lesson 4.3, Teacher Guide, Deep Dive, Lesson Planning Guidance: Day 1, “Starter Choice Board: Building Blocks or Blast from the Past (5-10 minutes) In this lesson, the ‘Building Blocks’ task asks students to access background knowledge on multiplying fractions. Use this activity if many of your students need support in recalling this skill. Consider using Expert Crayons to have students move around the room supporting each other. Choose the Starter Choice Board’s ‘Blast from the Past’ task to give students an opportunity to utilize problem solving skills involving percents and ratios.”

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for containing adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

Each Unit’s Planning & Assessment page includes a PD Library that provides teachers with access to Achieve the Core open-source publications from Student Achievement Partners. These documents offer adult-level explanations of mathematical content, organized by vertical progression within each domain. Additionally, the Planning & Assessment page contains a Unit Overview with the following information:

The Content Analysis section explains the major mathematical concepts taught in the unit, providing examples and explanations to enhance teachers’ understanding of both the content and its vertical progression within the standards. It also illustrates the types of tasks and procedures students will encounter. For example: 

• Unit 2, Planning & Assessment, Unit Overview, Content Analysis, states, “In this unit, students will build upon their previous understandings of fractions, multiplication and division to explore the concepts of ratios and rates. A ratio is a comparison between two or more quantities that may have the same units (four tennis balls to two golf balls) or different units (14 miles every two hours). Ratios involving different units are often referred to as rates. Students will use ratio language, such as ‘to,’ ‘per’ and ‘for every,’ to describe the relationship between quantities and they will represent ratio relationships in many forms to explore equivalence (or proportionality). These representations, which will be introduced early in the unit, include the following:” Visual student examples of Words & Numbers, Tape Diagrams, Graphs, Tables, and Double Number Lines are provided for teachers to review. “Toward the end of the unit, students will use ratio and rate concepts and representations to solve problems in authentic contexts, such as to make comparisons of speed, duration or price and to convert measurements within or between systems. For example, to convert from centimeters to meters, students could create a table of equivalent ratios or multiply the number of centimeters by the conversion factor $\frac{1m}{100cm}$. Later in the year, students will revisit ratios to develop an understanding of percentages (as a rate per 100). In future courses, students will expand their understanding of equivalent ratios to compute with ratios of rational numbers and to explore the algebraic nature of proportionality. For example, the ratio of two birch trees to five oak trees can be represented by an equation in the form $y=rx$, where r represents the unit rate. In this case, the equation $y=\frac{2}{5}x$ can be used to show $\frac{2}{5}$ birch tree, x, for every one oak tree, y. Explorations of proportionality will eventually connect to students’ work with geometry and statistics in Grades 7 and 8, and evolve into the study of linear and nonlinear functions in Grade 8 and beyond.”

The Learning Progression section explains and provides specific examples of the vertical progression of standards within the unit’s targeted domains. These examples include diagrams, models, numerical or algebraic representations, sample problems, and solution pathways. The Learning Progression is structured under the headings: ‘Previously, students have…, ‘In this unit, students will…,’ and ‘In the future, students will…’ with corresponding standards identified. For example:

• Unit 5, Planning & Assessment, Unit Overview, Content Analysis, Readiness Check & Learning Progression states, “In this unit, students will… Write, read and find the value of numerical expressions, including expressions with powers. 6.EE.A.1, Write, read and evaluate algebraic expressions. 6.EE.A.2, 6.EE.B.6, Identify and generate equivalent expressions by distributing and combining like terms. 6.EE.A.3-4, 6.NS.B.4”

• Unit 7, Planning & Assessment, Unit Overview, Content Analysis, Readiness Check & Learning Progression states, “In the future, students will… Add, subtract, multiply and divide rational numbers. 7.NS.A.1-3, Solve real world and mathematical problems involving inequalities and graph their solutions. 7.EE.B.4b, Understand and compare irrational numbers by approximating them to rational numbers. 8.NS.A.1-2” 

Each lesson’s Teacher Guide includes a Common Misconceptions section, which identifies common errors and provides explanations and recommendations to help students develop a stronger understanding. For example: 

• Unit 10, Lesson 10.2, Teacher Guide, Supports for Students with Learning and Language Differences, Common Misconceptions states, “Some students forget to divide the SUM when finding the mean and rather just divide the last number. Have students predict the mean of a set of numbers prior to calculating so that they can easily spot a miscalculation such as this. The prevalent misconception that range is a measure of center can be rectified by reinforcement of the distinction between center and variation. Some students describe the spread of data as low to high, e.g. 5-22. Remind them that the range (spread) is a single number, e.g., 17, and is used to describe the variation of the values across the data set. Underscore the purpose of the spread is not the value itself, but the interpretation it provides for the variation of the data. Students often have a hard time remembering which measure of center is which. Try helping them by giving them some visuals or word cues. The median can be associated with the word ‘middle’. Also, you might remind students that the yellow-painted line on a street is called the ‘median line’. The mode is ‘the most’ in that it is the number that appears the most. The mean can be explained as being ‘the mean one’ since it requires the most calculation to find.”

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Standards correlation information is included to support teachers in making connections from grade-level content to prior and future content. Standards can be found in multiple places throughout the course, including the Course Level, Unit Level, and Lesson Level of the program. Examples include:

• Each Unit’s Planning and Assessment section includes a Pacing Guide & Correlations, where the EdGems Math Course 1 Content Standards Alignment lists all grade-level standards along with the specific lessons where they are addressed. The program provides a structured approach to standards alignment through its Focus and Connecting Standards framework. A correlation chart is included, organizing standards into columns that indicate where each standard is taught as a Focus Standard in specific lessons and as a Connecting Standard across different units. This structure helps ensure that concepts are reinforced and revisited throughout the course.

  • “EdGems Math supports students’ proficiency in the Common Core State Standards through a program-design which supports the interconnectivity of mathematical ideas while providing clear learning objectives. This is achieved by designating Focus Standards in each lesson and Connecting Standards in each unit. The qualifiers of Focus and Connecting Standards were developed by the EdGems Math authoring team to design a scope and sequence in which mathematical ideas build upon each other and are revisited throughout the course. Each EdGems Math lesson identifies one or more standards as a Focus Standard to provide a focal point for the lesson objectives. The unit then provides opportunities for further connections to other standards across clusters and domains. These Connecting Standards offer opportunities for students to draw up and apply many mathematical ideas throughout the unit. The following chart shows when each standard is aligned as a Focus Standard or Connecting Standard throughout the course. Further explanations of the Focus and Connecting Standards are available within each Unit Overview.”

• Unit 8, Planning and Assessment, Unit Overview, Standards Correlation, Focus Content states, “The Focus Content Standard in this unit is itself a major cluster. This unit concludes focused instruction on all major clusters for the year, though the final units in this course will make meaningful connections to major clusters. This unit introduces students to two-variable equations in which students work with relationships between quantities. The unit focuses on representing independent and dependent variables using words, graphs, tables and equations. This unit lays the groundwork for continued explorations of linear relationships in future grades. The standard in this unit is formatively assessed throughout the unit and summatively assessed in the unit’s Test Prep, Performance Assessment and Unit Assessments.”

• Each Unit's Planning & Assessment section includes a PD Library with resources from Achieve the Core to support professional learning and instructional planning. These resources offer in-depth explanations of mathematical progressions aligned with the Common Core State Standards.

  • “CCSS Math Learning Progressions: Student Achievement Partners, a nonprofit organization, developed Achieve The Core to provide free professional learning and planning resources to teachers and districts across the country. The narrative documents below provide adult-level descriptions of the progression of mathematical ideas within domains or topics within the Common Core State standards for Mathematics.”

The Planning & Assessment sections within each unit provide coherence by summarizing content connections across grades. These sections highlight how mathematical concepts build upon prior knowledge and prepare students for future learning. Examples of where explanations of the role of specific grade-level mathematics appear in the context of the series include:

• Unit 1, Planning & Assessment, Unit Overview, Connecting Content Standards states, “In this unit, students are able to make introductory connections to many standards. For example, students may need to rewrite decimal division as an equivalent ratio of two whole numbers (6.RP.A.3a). Students also encounter situations involving decimal numbers where they must determine the unknown number in an expression or equation (6.EE.B.6). In these situations, number sense strategies will be used to determine the unknown number, which will provide a conceptual foundation for solving equations in a later unit (6.EE.B.7). Students will also apply decimal computation to real-world situations involving unit rates (6.RP.A.2, 6.RP.A.3) and areas and perimeters of polygons (6.G.A.1).”

• Unit 4, Planning & Assessment, Unit Overview, Readiness Check & Learning Progression includes a structured breakdown of prior learning, current learning, and future learning to reinforce coherence across grades. It states, “Previously, students have… Interpreted a fraction as a division problem (5.NF.B.3), multiplied whole numbers and fractions by fractions (5.NF.B.4), divided whole numbers by unit fractions and unit fractions by whole numbers (5.NF.B.7), and created equivalent decimal division expressions (6.NS.B.2-3). In this unit, students will… Divide fractions with models (6.NS.A.1), divide fractions using the standard algorithm (6.NS.A.1), and solve real-world problems by dividing fractions by fractions (6.NS.A.1). In the future, students will… Divide rational numbers (7.NS.A.2-3), compute unit rates from ratios with fractions (7.RP.A.1), and solve equations with rational number coefficients (7.EE.B.3).”

• Unit 7, Lesson 7.3, Teacher Guide, Standards Overview, Focus Content Standard(s): 6.EE.B.5 (Major), 6.EE.B.8 (Major) and Focus Math Practice Standard: SMP2. Starter Choice Board Overview states, “Storyboard: Order rational numbers (6.NS.C.7) Building Blocks: Determine which numbers make an inequality true (5.NBT.A.3) Blast from the Past: Perform ratio applications (6.RP.A.3) Fluency Board Skills: Solve one-step equations, simplify algebraic expressions, evaluate expressions.”

The materials reviewed for EdGems Math (2024) Grade 6 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 within each unit’s Planning & Assessment, About EdGems Math, Research Guide. 

Research Guide states, “Middle school is a critical stage for math instruction. Students form conclusions about their mathematical abilities, interests, and motivation.^1-10 Middle school students in the United States are falling behind compared to other countries in their math performance.² Studies have shown that struggles with math are particularly acute in middle school grades. The transition from elementary to middle school can lead to students falling behind with accumulated learning gaps.^3-5 Research shows that the mathematical achievement of middle schoolers has a direct impact on the likelihood that they will persist through the challenging material in pathways that can prepare them for the broadest range of options in high school and beyond.^6-7 Within this crucial time frame, a principal goal for middle school math teachers is to create a learning environment in which students are encouraged to see themselves as capable thinkers and doers of mathematics. Research demonstrates that to do this successfully, instructional materials must provide teachers with opportunities to 1) build upon and expand students’ cultural knowledge bases, identities, and experiences, 2) actively support students’ conceptual understanding, engagement, and motivation, 3) provide relevant, problem-oriented tasks that enables them to combine explicit instruction about key ideas with well-designed inquiry opportunities, and 4) spark student peer-to-peer discussion, perseverance and curiosity as they think and reason mathematically to solve problems in mathematical and real-world contexts.⁸ EdGems Math has been intentionally designed to support the diverse mathematical journeys of middle school students as they grow in their learning, critical thinking, and reasoning abilities. To reach the goal of higher order thinking for all, the EdGems Math curriculum connects each grade’s foundational math concepts to authentic, real-world contexts taught in multi-dimensional ways that meet a variety of learning needs. EdGems Math empowers teachers to adjust the content and instructional strategy and tailor outcomes of how learning is assessed.^9-10 EdGems Math curriculum is comprehensive, rigorous, and focused. It draws on decades of research exploring the best methods for teaching and learning math.”

The Unit Planning & Assessment, About EdGems Math, Research Guide incorporates multiple research-based strategies to support student learning:

• “Explore! Activities: The lesson-based Explore! Activities engage students in scaffolded tasks, guiding students as they begin grappling with the big ideas of the lesson and discovering new concepts (Small, M. and Lin, A.). The steps of the Explore! Activities move students through ‘Comprehension Checkpoints’ (National Council of Teachers of Mathematics) to guide information processing, ensure prior knowledge is activated, and discover patterns, big ideas, and relationships. Utilizing a student-centered approach, Explore! Activities engage students in the Standards of Mathematical Practice, which allows teachers to better facilitate learning using effective mathematical teaching practices (McCullum, W.). Every Explore! Activity provides students with ways to connect to key concepts through investigative, discovery-based tasks, culminating in an opportunity to generalize or transfer learning and move toward procedural and strategic proficiencies (California Department of Education).”

• “Lesson Presentation Communication Breaks: Communication Breaks are integrated into each Lesson Presentation as an opportunity for students to make sense of their learning. Each Lesson Presentation features two of seven structures to support students in the communication of their ideas or questions directly with their peers. The use of sentence stems in each Communication Break increases accessibility, enabling students to develop both social and academic language as they reflect on their learning (Smith et al.). The structures of Communication Breaks allow teachers to elicit student thinking, provide multiple entry points, focus students’ attention on structure, and facilitate student discourse (Chapin, S.H., O’Connor, C., Anderson, N.). As a result, students engage in the Standards of Mathematical Practice and gradually become more secure in their understanding and abilities to develop their knowledge (Bay-Williams, J.M., & Livers, S.).”

• “Mathematical Language Routines: Within every Teacher Lesson Guide are instructional supports and practices called Mathematical Language Routines to help teachers recognize and support students’ language development in the context of mathematical sense-making when planning and delivering lessons (Aguirre, J. M. & Bunch, G. C.). While Mathematical Language Routines can support all students when reading, writing, listening, conversing, and representing in math, they are particularly well-suited to meet the needs of linguistically and culturally diverse students. When students use language in ways that are purposeful and meaningful for themselves, they are motivated to attend to ways in which language can be both clarified and clarifying (Mondada, L. & Pekarek Doehler, S.). These routines help teachers ‘amplify, assess, and develop students’ language in math class’ (Zwiers, J. et al).”

• “Lesson Presentation: The editable Lesson Presentation enables teachers to shape lessons of balanced instruction on mathematical content and practice as they guide students through productive perseverance, small group instruction, and growth mindset activities. Components of the Lesson Presentation include Fluency Routines to develop number sense, vocabulary terms with definitions, examples with solution pathways, extra examples, the Explore! activity, Communication Breaks, and a formative assessment Exit Card. Teaching tips provide guidance on independent, group, and whole-class instruction. Studies identify explicit attention to concepts and students’ opportunity to struggle (as during the Explore! activity and Communication Breaks) as key teaching features that foster conceptual understanding (Hiebert, J., & Grouws, D. A.).”

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

Comprehensive lists of materials needed for instruction can be found in the PD Library link on the Unit Planning & Assessments page. The Required & Recommended Materials document provides a lesson-by-lesson breakdown of necessary resources for the course. Additionally, the Teacher Guide for each lesson includes a list of required materials. Examples include:

• Unit 1, Lesson 1.3, Teacher Guide, Materials states, “Required: Base ten blocks (for the Explore! activity)”

• Unit 6, Lesson 6.2, Teacher Guide, Materials states, “Required: Algebra tiles and equation mats (for the Explore! Activity)”

• Unit 10, Lesson 10.6, Teacher Guide, Materials states, “Required: Stacking blocks or cubes (for Explore! activity) Optional: Graph paper, colored pens/pencils, color tiles, unifix cubes, digital graphing software”

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

Each Unit Overview outlines standards alignment for formal assessments, including the Readiness Check, Performance Assessment, and Fluency Pre- and Post-Assessments. The EdGems website provides grade-level standards alignment for all formal assessments, including Assessments, Tiered Assessments, Performance Assessment, Unit Review (Print and Online), and Review & Assessments, accessible via the information (“i”) button at the bottom right corner of the icon. Each question's assessed standard(s) is listed for teachers, while only the Performance Assessment and Performance Task include practice standards. Examples include:

• Unit 1, Planning & Assessment, Unit Overview, Performance Task states, “In ‘Basketball Team,’ students will apply decimal operations to determine which uniforms the basketball team should purchase. Students will attend to precision (SMP6) and communicate stories with data as they compute pricing options involving decimals (6.NS.B.3) to make sense of costs and budgets.”

• Unit 1, Planning & Assessment, Unit Overview, Performance Assessment states, “In this Performance Assessment, students will evaluate the cost of designing and building an addition to an animal shelter. Students will reason abstractly and quantitatively (SMP2) as they explore changing quantities using multi-digit operations on whole numbers and decimals (6.NS.B.2-3), while considering the impact of local animal shelters in their communities.”

• Unit 5, Materials, Online Review & Assessments, Online Unit Review, Item 1 states, “What is the value of $10-8\div2+1$?” The assessment information identifies the standard alignment as 6.EE.A.1.

• Unit 9, Planning & Assessment, Assessments, Exercise 5 states, “A rectangular postcard has an area of 44 square inches. The width of the postcard is 8 inches. What is the postcard’s length?” The assessment information identifies the standard alignment as 6.G.A.1.

The materials reviewed for EdGems Math (2024) Grade 6 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. 

Each Lesson At A Glance in the Teacher Guide outlines specific learning goals, ensuring that teachers know the math standards that students should be able to demonstrate by the end of the lesson. Exit Cards serve as formative assessments, providing a real-time snapshot of student understanding, while a related Student Lesson exercise offers another checkpoint to identify students needing additional support. The EdGems website provides rubrics for scoring Exit Cards, ensuring consistency in evaluation.

The Online Class Results page generates automatic proficiency ratings based on student performance in Online Practice, Online Challenge, Test Prep, and Online Unit Reviews. These ratings align with state assessment benchmarks, helping teachers interpret mastery levels. Assessment Scoring Guides for Unit Assessments and Tiered Assessments follow the same ranking system, allowing teachers to track progress across multiple assessment formats.

Teachers receive real-time student performance insights through Teacher Gem activities, which embed informal assessment opportunities. For example, in Relay, teachers track how often a team revises an answer before moving forward, and in Ticket Time, class dot plots allow teachers to identify common errors. The PD Library provides written and video-based facilitation guides to support teachers in implementing these strategies effectively.

The Lesson Guide Deep Dive helps teachers analyze assessment data and adjust instruction accordingly. Exit Card results inform assignments for Leveled Practice and Differentiation Days, while the Differentiation Day guides provide self-assessments and targeted rotations to meet diverse learning needs. The Deep Dive section also identifies common misconceptions, equipping teachers with strategies to proactively address misunderstandings.

Performance assessments include rubrics for teacher grading and student self-reflection prompts, reinforcing the Standards for Mathematical Practice (SMP). The Unit Overview and Lesson Guide At A Glance ensure that all assessments and activities align with content and practice standards, with detailed mapping to Readiness Check skills, Storyboards, Performance Tasks, Fluency Boards, and Tiered Assessments.

To further support follow-up instruction, the Online Class Results tool provides recommended activities based on student proficiency levels, allowing teachers to tailor instructional strategies. By incorporating structured assessments, clear proficiency guidance, real-time monitoring tools, and differentiation strategies, EdGems Math ensures teachers have the necessary resources to assess student learning, interpret performance data, and provide targeted follow-up instruction.

Examples include: 

• Unit 1, Planning & Assessment, Performance Assessment, Performance Assessment Rubric ~ Student Reflection states, “Describe at least two ways you demonstrated the Focus Math practice below while completing this performance assessment. SMP2 I can reason about problems using words, numbers, symbols, and operations. I used numbers and symbols to represent an everyday situation. I explained what my answer meant in context of the problem. I understood which operations or strategies to use.” The Performance Assessment Rubric ~ Teacher Grading Rubric consists of four categories rated on a scale of 4 to 1 and a space for Comments. The four categories are: “Making Sense of the Problem: Interpret the concepts of the assessment and translate them into mathematics. Representing and Solving the Problem: Select an effective strategy that uses models, pictures, diagrams, and/or symbols to represent and solve problems. Communicating Reasoning: Effectively communicate mathematical reasoning and clearly use mathematical language. Accuracy: Solutions are correct and supported.”

• Unit 10, Planning & Assessment, includes Form A and Form B of the Unit 10 Assessment, along with an Answer Key and Assessment Scoring Guidelines for each question states, “3-point Items: #1, 2, 9, 12 Items that are each worth three points consist primarily of Depth of Knowledge Level 3 items considered ‘Strategic Thinking.’ Students may earn partial credit on items when showing progress on a solution pathway that connects to the concept being assessed with one or more errors. Students can earn either 0, 1, 2 or 3 points for items in this category. 0 points: An incorrect solution is given with no work or with work that does not show understanding of the concept. 1 point: Progress is made towards a correct solution, but multiple errors have been made. OR A correct solution is given with no supporting work or explanation. 2 points: Progress is made towards a correct solution, but one small error is made. OR A correct solution is given with partial supporting work provided. 3 points: The correct solution is given and is supported by necessary work or explanation. Total Points Possible: 27 Not Yet Met 0-16, Nearly Meets 17-18, Meets 19-24, Exceeds 25-27” Similar guidance is provided for 1 and 2 point assessment items.

The materials reviewed for EdGems Math (2024) Grade 6 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.

Assessments align with grade-level content and practice standards through various item types, including multiple-choice, short answer, extended response prompts, graphing, mistake analysis, and constructed-response items. They are available as downloadable PDFs for in-class printing and administration or can be completed through the online platform. Examples include:

• Unit 2, Planning & Assessment, Assessments, Form B, includes two items that demonstrate the full intent of 6.RP.3b. Exercise 11 states, “A 20-ounce jar of peanut butter costs $8.00. A 30-ounce jar costs $10.50. Which jar of peanut butter is cheaper per ounce?” Exercise 16 states, “Carl walks home from school every day at a rate of 6 miles per hour. It takes him 30 minutes to reach home. His sister, Leslie, runs home from the same school every day at a rate of 8 miles per hour. How many fewer minutes does it take Leslie to reach home from school each day compared to Carl?”

• Unit 5, Planning & Assessment, Performance Assessment, includes two items that demonstrate the full intent of 6.EE.2, 6.EE.4, 6.EE.6 and SMP7. The materials state, “1. The expression 3 times S subtracted from 1500 represents Sam’s and Sarah’s distance in yards from Hewitt Elementary School based on how many seconds (S) they have been riding their bikes together. a. Write an algebraic expression that represents Sam’s and Sarah’s distance from Hewitt Elementary School. b. Sam uses the expression $3(600-S)-300$ to represent their distance in yards from Hewitt Elementary School. Sarah uses the expression $2(S-750)+S$. Do either Sam or Sarah have an equivalent expression to your expression from part a? Justify your answer using words and mathematics. 2. Ivan rides his bike faster than Sam and Sarah. The expression 1500 − 4𝑆 represents Ivan’s distance in yards from Hewitt Elementary School based on how many seconds (S) he has been riding his bike. a. How far is Ivan from the school after 100 seconds? b. How far is Ivan from the school after 3 minutes? Show work that supports your answer. c. Sarah thinks Ivan will arrive at the school in less than 8 minutes. Do you agree or disagree? Justify your answer using words and mathematics.”

• Unit 7, Materials, Online Review & Assessments, Unit Assessment, Form B, and Unit Review include three items that demonstrate the full intent of 6.NS.7. The materials state, “Form B: 3. Determine whether each statement is TRUE or FALSE. The numbers -8, -6, -2, 3 are listed from least to greatest. Negative one-third is less than negative one. These numbers are listed from greatest to least: $-2\frac{1}{4}$, -3, $-3\frac{1}{2}$. Form B: 4. Which of the inequality statements below form a true statement? Select all that apply. A. $\left|-7\right|<\left|-8\right|$ B. $\left|4\right|> \left|5\right|$ C. $-\left|3\right|> \left|-6\right|$ D. $\left|-9\right|> \left|1\right|$ E. -$\left|5\right|<\left|1\right|$ F. $-\left|2\right|=\left|-2\right|$ Unit Review, 5. Marco recorded the four coldest outdoor temperatures from last year. The temperatures were $11^{\circ},9^{\circ},-13^{\circ}$, and $-8^{\circ}$. He ordered the temperatures from least to greatest: $-8^{\circ},9^{\circ},11^{\circ},-13^{\circ}$. Did he order them correctly? If not, correct his mistake.”

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

For each lesson, teacher guidance is provided alongside the Teacher Gem activity, which includes strategies to support instruction and student engagement. Printable PDF resources, such as the Student Lesson Textbook and Interactive consumable, include tools like graphic organizers, sentence stems, number lines, and coordinate planes. Lessons are also available as e-books with features including adjustable font sizes, text highlighting, text-to-speech, and note-taking tools.

Resources that support students in special populations to actively participate in learning grade level mathematics include:

• Differentiation Days: Differentiation Days are designed to provide teachers with structured opportunities to work with small groups based on specific learning targets. During these sessions, other students participate in mixed-ability group rotations, including the Teacher Small Group Rotation, Additional Practice Rotation, Application Rotation, and Tech Rotation. 

• Leveled Practice: The program includes three levels of leveled practice to address varying student needs. “Leveled Practice-T” is structured for students with learning and language differences, offering shorter problem sets, additional workspace, and simplified terminology and numbers to align with accessibility needs while maintaining grade-level alignment.

• ELL Supports: ELL supports are provided in the Planning and Assessment menu of each unit. These include explanations of Mathematical Language Routines (MLRs) and specific directions for incorporating these routines into lessons and activities. 

Each lesson includes Spanish translations of the student lesson, Explore! activities, leveled practice, and Exit tickets. Accompanying videos are included to guide students through the lesson content. These resources are structured to support student learning and accessibility.

Examples of the materials providing strategies and support for students in special populations include:

• Unit 4, Lesson 4.2, Lesson Presentation, Slide 14, Communication Break, Think, Ink, Pair, Square states, “When dividing fractions, is it possible to have a quotient be larger than the dividend? How about larger than the divisor? Think by yourself. Write down an idea. Share with a partner. Join with another partner set. We think… Do you agree or disagree? I respectfully agree/disagree because…” Teacher Guide, Lesson Presentation, “Have students utilize their interactive textbooks or composition notebooks to participate in guided note taking using the Lesson Presentation. Have students attempt Extra Examples with partners, in small groups or independently. Use ‘Communication Break’ slides as opportunities for meaningful discourse. Communication Break–Think, Ink, Pair, Square: Use the prompt “When dividing fractions, is it possible to have a quotient be larger than the dividend? How about larger than the divisor?” Have students think and write independently before joining with a partner to share. Then have two partner sets join together. Ask one group to start and the other group respond using the sentence stems provided.”

• Unit 10, Lesson 10.6, Teacher Gem, Masterpiece, Masterpiece Instructions states, “NOTE: Since the first task card often has the lowest math level, the teacher may choose to give this task card to the student who has shown the lowest skills in the given concept as a scaffolding technique. Note: If some partner sets are falling behind other groups, you may designate ‘Expert Groups’ that are further along. These groups have the responsibility of stopping what they are doing when another group approaches and asks for help. Expert groups should be chosen carefully and consist of students you trust to help rather than ‘give away the answers.’”

The materials reviewed for EdGems Math (2024) Grade 6 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

Each unit includes multiple opportunities for students to engage with grade-level mathematics at increasing levels of complexity. These opportunities are embedded within lessons and available to all students, supporting a range of learners in exploring mathematical concepts at higher levels of complexity.

Several resources and features within the program provide opportunities for extended mathematical exploration.

• Performance Task: “The Performance Task provides applications of most or all of the standards addressed in the unit. This task contains Depth of Knowledge Level 3 and 4 strategic and extended thinking questions where students apply multiple standards in a non-routine manner to solve. These tasks provide entry points for all levels of learners and encourage students to explain their thought processes or critique the reasoning of others.”

• Performance Assessment: These non-routine problems require students to engage in higher-level thinking while applying their knowledge of the standards.

• Leveled Practice: The “C” in each lesson is designed for students who have already demonstrated proficiency. It extends their learning by making connections to future standards and incorporating Depth of Knowledge (DOK) Level 3 or 4 exercises.

•  Tic-Tac-Toe Boards: According to the EdGems Math Program Components, “Each Tic-Tac-Toe Board includes nine activities that extend or look at the content of the unit in different ways. The Tic-Tac-Toe Boards include activities that make use of a variety of multiple intelligences.”

• Teacher Gems: Teacher Gems include problem sets at multiple levels of complexity, allowing for differentiated problem-solving experiences. For example, activities such as Four Corners, Relay, and Stations include multiple levels of complexity within the tasks.

• Online Practice & Exit Card Resource: This resource offers five options for each lesson: Two Online Practice sets (A and B), each containing six items at the proficient level. Two Online Challenge sets (A and B), each containing four challenge questions. Attempt A provides immediate feedback on correctness, while Attempt B includes worked-out solution pathways to help students identify errors in their work.

The materials include structured activities that provide opportunities for students to engage with mathematical concepts at increasing levels of complexity.

• Unit 3, Materials, Tic-Tac-Toe, Sales Tax states, “Most states have a sales tax which is applied to non-grocery items. If a state has an 8% sales tax, you pay the cost of the item and an additional 8% of the item’s cost. Example: An item costs $100. An 8% sales tax is added to the item cost. You pay $108 total. Step 1: Research and list the states in the United States that do not charge sales tax. Step 2: Find the two states which have the highest sales tax. Step 3: Find the two states which have the lowest sales tax greater than zero. Step 4: Suppose you want to buy each of the items listed below. Write each item’s total cost (including sales tax) if you purchase it in (1) your state, (2) the state with the highest sales tax and (3) the state with the lowest sales tax. Laptop computer for $800. Used car for $6,000. Television for $1,500.”

• Unit 6, Lesson 6.4, Teacher Gem, Stations, Directions state, “Print one or two sets of Station Cards. One set of Station Cards, numbered 1 through 8, can be for students who are still needing assistance at the entry level of the standard. A challenge set of Station Cards, lettered A through H, can be used to ask students to extend and apply their thinking around the standard. If using two levels of cards, printing the cards on two different colors of paper is advised (e.g. Cards 1-8 in one color and Cards A-H another color).” Station 1, “Find the value that goes into the box. 30% of 60 is ___.” Station A, “Yolanda ordered 40 pieces of new furniture for her showroom floor. Of the furniture ordered, 20% were recliners. How many recliners did Yolanda order?”

The materials reviewed for EdGems Math (2024) Grade 6 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.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the ELL Supports Guide, “We view the background knowledge, experiences, and insights that English Learners bring to the classroom as strengths to be leveraged, and we are committed to ensuring that they receive academic success with rigorous grade-level curriculum. In recognition of the unique needs of learners, including those with diverse levels of mathematical proficiency, our curriculum includes research-based guidance for differentiated English Language Learner (ELL) instruction."

• The ELL Supports Guide outlines strategies for students who read, write, and/or speak in a language other than English to engage with grade-level mathematics. Key areas of focus include scaffolding tasks, fostering mathematical discourse, and incorporating instructional strategies informed by research. Tasks include scaffolds and language supports designed to facilitate mathematical understanding. The instructional design integrates opportunities for students to express their mathematical thinking both orally and in writing.

• The ELL Supports Guide contain recommendations related to student assessments. Additional resources in the materials include Target Trackers and Math Practice Trackers, which align with structured conferencing planned three times per unit. A Math Self-Assessment Rubric is included to support student reflection, along with a Sample Vocabulary Journal Format that provides space for root words, home-language translations, definitions, images, and sentence frames.

• Each lesson’s Teacher Guide includes three lesson-specific Mathematical Language Routines (MLRs), with two MLRs suggested for implementation per lesson. Strategies described in the materials include language modeling through think-alouds, the use of visual aids featuring key vocabulary, and a multilingual glossary with online vocabulary available in ten languages. Videos within the ELL Supports Guide provide examples of teachers breaking down tasks, using cognates, and prompting students to explain their thinking. Language functions are also included to structure discussions.

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

• Unit 1 Lesson 1.2, Teacher Guide, Lesson Presentation states, “Have students utilize their interactive textbooks or composition notebooks to participate in guided note taking using the Lesson Presentation. Have students attempt Extra Examples with partners, in small groups or independently. Use ‘Communication Break’ slides as opportunities for meaningful discourse. Communication Break – Estimation Pause: Use an ‘Estimation Pause’ for Example 2. Have students work with partners or small groups to complete one of the sentence frames provided on the slide and be ready to share their reasoning. Then work together to solve the task and compare to initial estimates.” Lesson Presentation Example 2 states, “Communication Break – Estimation Pause 1. Examine the problem. 2. Without writing, estimate the answer or range of answers. I think the answer will be between and because… I think the answer will be more/less than because… I think the answer may be about because.” “Communication Break – Silent Teacher: Use the ‘Silent Teacher’ strategy with the provided video or your own work to have students observe and process Example 3.” Lesson Presentation, Example 3, states, “Communication Break – Silent Teacher 1. Watch your teacher work through the problem. 2. Discuss with your partner or group how the problem was solved. First, they , because… Then, they , because… ; I wonder why they… ; I didn’t understand why they… ; I recognize…” A video link is provided for students to watch. 

• Unit 2, Lesson 2.3, Teacher Guide, Supports for Students with Learning and Language Differences, Common Misconceptions states, “Students may struggle with setting up a rate from an application situation by mixing up the numerator and denominator. Have students reason about what rate makes more sense (i.e., reading the rate with the units each way like miles per hour or hours per mile). Students may struggle to see a unit rate if the denominator does not have a 1 written (i.e., $9 per ticket). If no numerical value is listed, help students understand that this is the same as one (i.e., $9 per 1 ticket). Some students may identify unit rates as a ratio in lowest terms or simplest form. Teachers should make explicit the difference between ‘lowest terms’ and ‘simplest forms’ with ‘one unit’. Teachers can compare two rates that can be simplified, with only one being able to be called a unit rate. Teachers can also model unit rates that can’t be easily reduced, and provide additional opportunities for students to practice identifying and writing unit rates. Some students have difficulty understanding unit rates. Using pictures or diagrams may help students to better visualize a problem. This might include using tape diagrams to show multiples of the unit rate next to shaded clocks representing the amount of time that has passed.”

• Unit 6, Lesson 6.1, Teacher Guide, Supports for Students with Learning and Language Differences, Mathematical Language Routines states, “MLR 8 – Discussion Supports: As students work through the Teacher Gem activity Partner Math, ask students to share their thinking aloud with their partners. Provide prompts for students to ask each other clarifying questions (i.e. ‘How do you know _ is/is not a solution?’). As you listen to explanations from different groups, ask students to repeat their reasoning with more precise mathematical language (i.e. ‘Can you restate that using the word solution?’)”

The materials reviewed for EdGems Grade 6 meet expectations for providing manipulatives, physical but not virtual, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Materials consistently include suggestions and links to manipulatives to support grade-level math concepts. The Teacher and Student Moves for Math Practice 5 and Explore! Activities incorporate physical manipulatives when appropriate, with required materials listed in the full course materials list under the Teacher Guide At A Glance section. Student Gems in each lesson provide virtual manipulatives, such as Desmos and Geogebra, to help students make sense of concepts and procedures. Examples include:

• Unit 1, Lesson 1.3, Teacher Guide, Explore! Activity: Beaded Necklaces states, “This Explore! activity is designed to build conceptual understanding of division using base-ten blocks (a tool commonly used in elementary grades). If base-ten blocks are not available, students can use grid paper or templates from the internet. The first page models the division process using base-ten blocks, then students apply this process to solve the steps on the backside of the activity sheet. The process of using base-ten blocks to model division allows for students to physically show how the values in each place value are taken apart and put back together to find the quotient. Implementation Option #1: Teachers may choose to facilitate this activity as a full class using a document camera if supplies are not readily available or if teachers are short on time. Teachers would model the first page with base-ten blocks for students. Then, for each of the five parts of Step 1, the teacher can call an individual or partner set up to the front of the classroom. The rest of the class would verbally explain the process while the individual or partner set models the process using base-ten blocks. Use Step 2 as an extension if time allows. Implementation Option #2: Give pairs of students a set of base ten blocks. Prior to using the activity sheet, ask students how they might model 52 divided by 4. Then have students get out their Explore! sheet and compare their strategy to the one shown. Have students work together to complete Steps 1-2.”

• Unit 4, Lesson 4.4, Student Gems, Geogebra states, “Bring mixed number multiplication to life with this interactive simulation.” Students enter their own multiplication problem, and a visual representation of the problem is displayed to strengthen understanding.

• Unit 10, Lesson 10.6, Teacher Guide, Math Practices: Teacher and Student Moves, SMP5 Teacher Moves states, “In this lesson, provide students with graph paper, colored pencils, color tiles, unifix cubes, digital graphing software, etc. with which students can practice and become comfortable using.”