6th Grade - Gateway 1

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Summative assessments provided by the materials include Chapter Tests, Cumulative Reviews, and Benchmark Assessments and are available in print and digitally. According to the Preface of the Math in Focus: Assessment Guide, "Assessments are flexible, teachers are free to decide how to use them with their students. ... Recommended scoring rubrics are also provided for some short answer and all constructed response items to aid teachers in their marking." The following evidence is based upon the provided assessments and acknowledges the flexibility teachers have in administering them in order to understand their students' learning.

The provided assessments, found in the Assessment Guide Teacher Edition, assess grade-level standards. Examples include:

• In Chapter Test 1, Section B, Item 10 states, “What is the value of 12^3 - 4^2 × 9^2?” (6.EE.1)

• In Cumulative Review 1, Section A, Item 8 states, “What is the value of 975 ÷ 0.78? A. 12,500, B. 1,250, C. 125, D. 12.5.” (6.NS.2)

• In the Mid-Year Benchmark Assessment, Section B, Item 30 states, “A florist arranged red, pink, and white roses together into bouquets of the same size. The ratio of the number of red roses to the number of pink roses to the number of white roses is the same for all the bouquets. The table shows the different number of red, pink, and white roses in bouquets. Write each missing value in the table.” (6.RP.1)

• In Chapter Test 11, Section A, Item 2 states, “A cube has edges measuring 8 centimeters each. What is its surface area in square centimeters? A.192, B.256, C.384, D. 512.” (6.G.4)

• In Chapter Test 13, Section B, Item 7 states, “The data set shows the masses in grams of eight cell phones. 165, 93, 140, 185, 174, 143, 191, 122 Find the mean absolute deviation of the data set. Write your answer in the space below.” (6.SP.5c)

The provided assessments also assess above-grade assessment items that could be removed or modified without impacting the structure or intent of the materials. Examples include:

• In Chapter Test 1, Section C, Item 11 states, “Ann says the square root of 15^2 + 6^3 × 2^2 is 42. She explains that to find the answer, she first adds 15² and 6³. Then, she multiplies the sum by 2^2. Finally she finds the square root of the product. Explain why Ann’s reasoning is incorrect. Determine the square root of 15^2 + 6^3 × 2^2.” This item assesses 8.EE.2 (Evaluate square roots of small perfect squares and cube roots of small perfect cubes). 

• In Cumulative Review 1, Section C, Item 21, Part B states, “This question has three parts. The temperature at which a substance melts is called its melting point. The table shows the melting points of some elements. Name a pair of elements such that their melting points differ by 10°C.” A table with the list of elements is provided including Hydrogen, Oxygen, Nitrogen, Neon, and Flouride, with the respective melting points ($$\degree$$C) of −259, −219, −210, −249, −220.” This item assesses 7.NS.1 (Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers).

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Materials provide opportunities for students to engage in grade-level problems during Engage, Learn, Think, Try, Activity, and Independent Practice portions of the lesson. Engage activities present an inquiry task that encourages mathematical connections. Learn activities are teacher-facilitated inquiry problems that explore new concepts. Think activities provide problems that stimulate critical thinking and creative solutions. Try activities are guided practice opportunities to reinforce new learning. Activity problems reinforce learning concepts while students work with a partner or small group. Independent Practice problems help students consolidate their learning and provide teachers information to form small group differentiation learning groups.

The materials provide one or more Focus Cycles of Engage, Learn, Try activities and opportunities for Independent Practice which provide students extensive work with grade-level problems to meet the full intent of grade-level standards. Examples include:

• In Section 3.1, Dividing Fractions, students divide a fraction, whole number, or mixed number by a fraction or mixed number. In the Engage activity on page 93, students use spatial reasoning, manipulatives and drawings to build the conceptual understanding of dividing proper fractions by unit fractions. The activity states, “Show how you find the number of eighths in \frac{3}{4} using manipulatives or a picture. Now, find the number of thirds in \frac{5}{6}. Draw a picture to explain your thinking.” In the Learn activity, Problem 1, page 93, students divide a proper fraction by a unit fraction. The problem states, “Vijay had \frac{2}{3} of a pizza. He cut it into equal slices. Each slice was \frac{1}{6} of the whole pizza. How many equal slices did Vijay cut?” In the Try activity, Problem 1, page 98, students practice dividing a proper fraction by a proper fraction. The problem states, “David had 89 of a fruit tart and some plates. He put \frac{2}{9} of the fruit tart on each plate. How many plates did he have?” In Independent Practice, Problem 17, students practice dividing proper fractions by improper fractions. The problem states, “ \frac{1}{9} ÷ \frac{14}{3}”. Students engage with extensive work and full intent of 6.NS.1 (Apply and extend previous understandings of multiplication and division to divide fractions). 

• In Section 4.2, Equivalent Ratios, students write, simplify, and compare ratios. In the Engage activity on page 187, students discuss strategies for generating equivalent ratios. The activity states, “Look at the ratio 18:24. Can both terms be divided by 3? Can both terms be divided by 5? Make a list of numbers that both terms can be divided by. What do you notice about the numbers in the list? Now divide each term in the ratio 18:24 by the numbers in the list and write each result as a ratio. What can you say about the ratios?” In the Learn activity, Problem 1b, page 192, students work with descriptions of ratios to find quantities. The problem states, “To make 1 portion of dough, Sophia mixes 5 cups of flour with every 3 cups of water. Sophia wants to make 5 portions of dough. How many cups of flour and how many cups of water does she need?” In the Try activity, Problem 1, page 191, students practice working with tables of ratios. The problem states, “Mr. Smith uses the following table to prepare four mixtures of cement and sand using identical pails.” The table includes information about the number of pails of cement and sand for four different mixtures, and students are asked to identify the ratios of cement:sand, cement: sand (simplest form) and number of pails of cement/number of pails of sand as a fraction. In Independent Practice, Problem 39b, students practice working with descriptions of ratios to find quantities. The problem states, “Daniel uses 5 fluid ounces of lemonade concentrate for every 9 fluid ounces of orange juice concentrate to make a serving of fruit punch. If Daniel wants to make 4 servings of fruit punch, how many fluid ounces of lemonade concentrate and how many fluid ounces of orange juice concentrate does he need?” Students engage with extensive work and full intent of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems).

• In Section 7.1, Using Letters to Represent Numbers, students use letters to represent unknown numbers and write algebraic expressions. In the Engage activity on page 5, students use a variable to represent an unknown or missing value with a variable in an addition scenario. The activity states, “Lily buys some plants. Her friend gives her 2 more plants. How many plants does Lily have now? Explain your thinking. How do you express the number of plants Lily has now? Discuss.” In Learn activity, Problem 2b, page 7, students write algebraic expressions with subtraction. The problem states, “Subtract 3 from a. a - 3.” In the Try activity, Problem 4, page 8, students practice writing algebraic expressions with subtraction, “10 less than h.” In Independent Practice, Problem 11a, students practice writing algebraic expressions with the four operations. The problem states, “Bianca is now x years old. Her father is 24 years older than her. Find her father’s age in terms of x.” Students engage with extensive work and full intent of 6.EE.2 (Write, read, and evaluate expressions in which letters stand for numbers).

• In Section 11.3, Volume of Rectangular Prisms, students find the volume of rectangular prisms. In the Engage activity, page 237, students calculate the volume of a rectangular prism with a fractional side length. The activity states, “Explain the steps you would take to find the volume of a rectangular prism that measures 4 centimeters by 6 centimeters by 3\frac{1}{2} centimeters. Compare your method to your partner’s.” In the Learn Activity, Problem 3, page 238, students calculate the volume of a rectangular prism. The problem states, “A rectangular prism measures 8.4 centimeters by 5.5 centimeters by 9 centimeters. What is its volume?” In the Try activity, Problem 1, page 240, students practice finding the volume of a rectangular prism. The problem states, “Find the volume of each rectangular prism. Length = 6 in. Width = 5\frac{1}{4} in. Height = 12 in.” In Independent Practice, Problem 8, students practice finding the number of cubes required to fill a rectangular box completely. The problem states, “Cameron fills the rectangular box on the right with \frac{1}{2}-inch cubes completely. How many \frac{1}{2}-inch cubes does he need?” The box shown on the right has dimensions of 15 in., 9 in., and 6 in. Students engage with extensive work and full intent of 6.G.2 (Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism).

• In Section 12.1, Collecting and Tabulating Data, students engage with extensive work and full intent of 6.SP.1 (Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answer). In the Engage activity, page 267, students explore strategies for data collection. The activity states, “Suppose you want to find out how many different types of vehicles pass by your school from 8 A.M. to noon on weekdays. How do you find out? List the steps you would follow and share your idea.” In the Learn activity, Method 2, page 267, students collect and tabulate data to answer a statistical question. The activity states, “Suppose you want to find out how your classmates get to school every morning. To find out, you can collect data by using one of the three methods. You can interview your classmates to find out how they get to school.” In the Try activity, Problem 3, page 270, students practice collecting and tabulating data to answer a statistical question. The problem states, “Decide if each sentence is a statistical question. Answer Yes or No. What is the average temperature in April?” In Independent Practice, Problem 5, students collect, organize, and tabulate data. The problem states, “Decide if each sentence is a statistical question. Answer Yes or No. How many hours do the sixth-graders volunteer in a year?” Students engage with extensive work and full intent of 6.SP.1 (Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answer).  

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations that, when implemented as designed, the majority of the materials address the major work of each grade.

• There are 13 chapters, of which 8 address major work of the grade, or supporting work connected to major work of the grade, approximately 62%.

• There are 53 sections (lessons), of which 35.5 address major work of the grade, or supporting work connected to major work of the grade, approximately 67%.

• There are 158 days of instruction, of which 104.5 days address major work of the grade, or supporting work connected to the major work of the grade, approximately 66%.

A day-level analysis is most representative of the instructional materials because the days include all instructional learning components. As a result, approximately 66% of the instructional materials focus on major work of the grade.

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Digital materials provide “Math in Focus 2020 Comprehensive Alignment to CCSS: Course 1” located under Discover, Planning. This document identifies the standards taught in each chapter’s section allowing connections between supporting and major work to be seen. Examples include: 

• In Section 1.2, Common Factors and Multiples, Try, Problem 3, page 15, students use the greatest common factor with the distributive property which connects the supporting work of 6.NS.4 (Find the greatest common factor of two whole numbers less than or equal to 100) with the major work of 6.EE.3 (Apply the properties of operations to generate equivalent equations). Students solve, “Express the sum of each pair of numbers as a product of the greatest common factor of the numbers and another sum. 60 + 85.”

• In Section 5.3, Distance and Speed, Try, Problem 2, page 263, students divide fractions to find unit rate which connects the supporting work of 6.NS.2 (Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions) with the major work of 6.RP.2 (Understand the concept of unit rate). Students solve, “A scooter travels $$\frac{1}{4}$$ mile in $$\frac{1}{2}$$ minute. Find the speed of the scooter in miles per minute.”

• In Section 7.4, Try, Problem 8, page 32, Expanding and Factoring Algebraic Expressions, students expand and factor simple algebraic expressions which connects the supporting work of 6.NS.4 (Use the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor) with the major work of 6.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions). Students solve, “Factor each expression. 12t - 8.”

• In Section 8.1, Solving Algebraic Equations, Independent Practice, Problem 21, page 66, students solve equations with one variable which connects the supporting work of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation) with the major work of 6.EE.7 (Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers). Students solve, “9.9 = x + 5.4.”

• In the Chapter 9 Review, The Coordinate Plane, Problem 20, page 149, students plot points on the coordinate plane which connects the supporting work of 6.G.3 (Draw polygons in the coordinate plane given coordinates for the vertices) with the major work of 6.NS.6b (Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane). Students solve, “Simone knows how to plot Point A at (-4, 3) on a coordinate plane. She needs to plot Point B at (-4, -3). Where is point B located on the coordinate plane in relation to Point A?”

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

Digital materials provide “Math in Focus 2020 Comprehensive Alignment to CCSS: Course 1” found under Discover, Planning. This document identifies the standards taught in each chapter’s section showing connections between supporting to supporting work and major to major work. 

There are connections from supporting work to supporting work throughout the grade-level materials, when appropriate. Examples include:

• In Section 10.1, Areas of Triangles, Independent Practice, Problem 8, students connect the supporting work of 6.G.A (Solve real-world and mathematical problems involving area, surface area, and volume) to the supporting work of 6.NS.B (Compute fluently with multi-digit numbers and find common factors and multiples). Students solve, “Find the area of each triangle.” A triangle with a height of 6 cm and base of 15.5 cm is pictured. 

• In Section 12.2, Dot Plots, Activity, Problem 3, page 277, students connect the supporting work of 6.SP.A (Develop understanding of statistical variability) to the supporting work of 6.SP.B (Summarize and describe distributions). Students toss two dice 20 times and record their results then, “Draw a dot plot to represent the data.” 

• In Section 13.4, Interpreting Quartiles and Interquartile Ranges, Independent Practice, Problem 4, students connect the supporting work of 6.NS.B (Compute fluently with the multi-digit numbers and find common factors and multiples) to the supporting work of 6.SP.B (Summarize and describe distributions). Students solve, “Find the median, the lower quartile, the upper quartile, and the interquartile range of each data set. Height, in feet, of 14 lemon trees: 12.4, 11.8, 14.6, 13, 11.2, 15, 13.4, 11, 12.8, 13, 11.2, 14.4, 12, and 13.2?”

There are connections from major work to major work throughout the grade-level materials, when appropriate. Examples include:

• In Section 5.3, Distance and Speed, Try, Problem 1, page 265, connects the major work of 6.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions) to the major work of 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems). Students find the distance given the speed and time using the formula distance = speedtime. Students solve, “A train can travel 120 kilometers per hour. How far can the train travel in 5 minutes?”

• In Section 8.1, Solving Algebraic Equations, Independent Practice, Problem 19 connects the major work of 6.EE.B reason about and solve one-variable equations and inequalities to the major work of 6.NS.A, apply and extend previous understandings of multiplication and division to divide fractions by fractions. Students solve, “Solve each equation using the concept of balancing. Express each answer in simplest form. $$\frac{8}{9} =\frac{1}{3}$$f”

• In Section 8.2, Writing Linear Equations, Independent Practice, Problem 7, connects the major work of 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems) to the major work of 6.EE.C (Represent and analyze quantitative relationships between dependent and independent variables). Students solve, “Use graph paper. Solve. There are x sparrows in a tree. There are 50 sparrows on the ground beneath the tree. Let y represent the total number of sparrows in the tree and on the ground. a. Express y in terms of x. b. Make a table to show the relationship between y and x. Use values of x = 10, 20, 30, 40, and 50 in your table. c. Graph the relationship between y and x on a coordinate plane.”