6th Grade - Gateway 2

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

Students have opportunities to develop procedural skill and fluency during the Engage and Learn portions of the lessons. Examples include:

• In Section 1.3, Squares and Cubes, Learn, Problem 2, page 23, students use the idea of area of squares to find the square of a whole number. The problem states, “Find the square of 5. $$5^2$$ = 5 × 5 = 25. The square of 5 is 25.” Students develop procedural skill and fluency of 6.EE.1 (Write and evaluate numerical expressions involving whole-number exponents).

• In Section 3.3, Adding and Subtracting Decimals Fluently, Engage, page 121, students recall prior knowledge of decimal addition. The materials state, “Recall and Discuss what you learned about place value and decimal addition. Show how you find the sum of 4.25 and 5.798. Show your method.” Students develop procedural skill and fluency of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation).

• In Section 3.5, Dividing Decimals Fluently, Learn, page 139, Method 2, students evaluate division expressions by expressing them as a fraction, then divide. The materials state, “Express the quotient as a fraction then divide. 0.56 ÷ 0.04 = $$\frac{0.56}{0.04}$$ = $$\frac{56}{4}$$ = 14. Express the quotient as fraction. Multiply both the numerator and denominator by 100 to make the divisor a whole number. Simplify.” Students develop procedural skill and fluency of 6.NS.2 (Fluently divide multi-digit numbers using the standard algorithm).

• In Section 6.1, Understanding Percent, Learn, Problem 2, page 308, students express a part of a whole as a fraction and a percent. The problem states, “72 out of 100 cats are long-haired cats. What percent of the cats are long-haired cats? 72 out of 100 → $$\frac{72}{100}$$ (Express the fraction as a percent.) 72% of the cats are long-haired cats.” Students develop procedural skill and fluency of 6.RP.3c (Find a percent of a quantity as a rate per 100).

• Lesson 9.2, Lengths of Line Segments, Engage, page 123, students find lengths of line segments on the x-axis and the y-axis. The materials state, “Plot Points A(2, 0), B(4, 0), C(0, 4) on the coordinate plane below. How many units along the x-axis is Point B from Point A? How many units along the y-axis is Point D from Point C?” Students develop procedural skill and fluency of 6.G.3 (Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate).

Students have opportunities to independently practice procedural skill and fluency during the Independent Practice portion of the lesson. Examples include: 

• In Section 3.4, Multiplying Decimal Fluently, Independent Practice, Problem 25, page 134, students multiply a multi-digit decimal by a number with one decimal place in vertical form. The materials state, “Write in vertical form. Then, multiply. 1.2 × 0.6”. Students independently practice procedural skill and fluency of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation).

• In Section 6.1, Understanding Percent, Independent Practice, Problem 4, page 313, students express each percent as a decimal. The materials state, “There are 750 spectators in the stadium, of which 420 are adults and the rest are children. a. What percent of the spectators are adults? b. What percent of the spectators are children?” Students independently practice procedural skill and fluency of 6.RP.3c (Find a percent of a quantity as a rate per 100).

• In Lesson 9.2, Lengths of Line Segments, Independent Practice, Problem 4, page 133, students find the lengths of line segments on the coordinate plane. The materials state, “Use graph paper. Plot each pair of points on a coordinate plane. Connect the points to form a line segment and find its length. G(-6, -3) and H(-6, -8).” Students independently practice procedural skill and fluency of 6.G.3 (Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate).

• In Section 13.4, Interpreting Quartiles and Interquartile Range, Independent Practice, Problem 2, page 345, students find the median, lower quartile, upper quartile, and interquartile range of each data set. The problem states, “Find the median, the lower quartile, the upper quartile, and the interquartile range of each data set. Scores of nine football players in a season: 33, 42, 31, 27, 47, 23, 40, 45, and 43.” Students independently practice procedural skill and fluency of 6.SP.5c (Giving quartile measures of center and variability).

Course 1 materials contain a separate Fact Fluency book so students can independently practice specific strategies to promote procedural skill and fluency. Examples include:

• In Fact Fluency, Chapter 2, Whole Numbers: Multiplication and Division, Problem 8, page 8, students use the standard algorithm to divide “454 ÷ 2 = ____.” Students independently practice procedural skill and fluency of 6.NS.2 (Fluently divide multi-digit numbers using the standard algorithm).

• In Fact Fluency, Chapter 3, Fractions and Decimals, Problem 9, page 21, students divide multi-digit decimals, “0.042 ÷ 0.03.” Students independently practice procedural skill and fluency of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation). 

• In Fact Fluency, Chapter 8, Equations and Inequalities, Problem 4, page 6, students solve algebraic inequalities by graphing. The problem states, “Graph the inequality on a number line to show the possible solutions. Then, test a solution. p -1.5.” Students independently practice procedural skill and fluency of 6.EE.8 (Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams).

• In Fact Fluency, Chapter 11, Surface Area and Volume of Solids, Problem 6, page 85, students use order of operations to solve, “$$2^2$$ × 9 ÷ 3 - 3 = _____.” Students independently practice procedural skill and fluency of 6.EE.1 (Write and evaluate numerical expressions involving whole-number exponents).

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Students have opportunities throughout the materials to engage in routine application of mathematics. Examples include:

• In Section 3.6, Real-World Problems: Decimals, Independent Practice, Problem 5, Page 148, students divide decimals to solve real-world problems. The problem states, “Maya buys 6.93 pounds of raisins to make some loaves of raisin bread. Each loaf requires 0.33 pound of raisins. How many loaves of bread can she make?” Students independently engage in routine application of 6.NS.3 (Fluently add, subtract, multiply and divide multi-digit decimals using the standard algorithm).

• In Section 4.3, Real-World Problems: Ratios, Independent Practice, Problem 7, page 212, students solve real-world problems involving two sets of ratios. The problem states, “The ratio of the number of mystery books to the number of science fiction books in a bookcase is 4 : 3. The ratio of the number of science fiction books to the number of biographies is 4 : 5. If there are 48 science fiction books, find the total number of books in the bookcase.” Students independently engage in routine application of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems).

• In Section 5.5, Real-World Problems: Speed and Average Speed, Learn, Problem 1, page 277, students draw and use diagrams to solve real-world problems involving speed, distance, and time. The problem states, “Adam cycled from Town X to Town Y. He started his journey from Town X at 7:45 A.M. and ended at Town Y at 9:15 A.M. He cycled at an average speed of 14 kilometers per hour. What distance did Adam cycle?” Students engage in routine application of 6.RP.3 (Students use ratio and rate reasoning to solve real-world and mathematical problems).

• In Section 7.5, Real-World Problems: Algebraic Expressions, Learn, Problem 1, page 35, students solve real-world problems involving algebraic expressions. The problem states, “Emma has y books. Matthew has 3 times as many books as Emma. Matthew buys another 7 books. How many more books does Matthew have than Emma? Give your answer in terms of y in the simplest form. If Emma has 25 books, how many more books does Matthew have than Emma?” Students engage in routine application of 6.EE.6 (Use variables to represent numbers and write expressions when solving a real-world or mathematical problem).

Students have opportunities throughout the materials to engage in non-routine application of mathematics. Examples include:

• In Chapter 4, Performance Task, Problem 3, page 228, students adjust their models and thinking to respond to changes in the given information. The problem states, “The sixth-grade raised $$\frac{2}{3}$$ as much money as the seventh-graders at the school fair in the morning. In the afternoon, the sixth-graders continued to raise more money. In the end, they raised twice as much money as the seventh- graders. Both the sixth-graders and seventh-graders raised $324 in the end. How much more did the sixth-graders raise in the afternoon? Show and explain your work.” Students independently engage in non-routine application of 6.RP.3 (Use ratio and rate reasoning to solve real-world problems).

• In Chapter 10, Put On Your thinking Cap! Problem 1, page 202, students find the area of a figure. The problem states, “Figure ABCD is made up of Square PQRS and four identical triangles. The area of Triangle APD is 49 square feet. The lengths of $$\bar{AP}$$ and $$\bar{PD}$$ are in the ratio 1 : 2. Find the area of Figure ABCD.” A diagram is provided. Students independently engage in non-routine application of 6.G.1 (Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes).

• In Section 11.3, Volume of Rectangular Prisms, Independent Practice, Problem 6, page 241, students find the length of an edge of a cube given its volume. The problem states, “Solve. A cube has a volume of 125 cubic inches. Find the length of its edge.” Students engage in non-routine application of 6.EE.2c (Evaluate expressions at specific values of their variables).

• In Section 13.1 Mean, Engage, page 318, students explore what they know about mean and how it relates to total and missing values. The materials state, “The mean of four numbers is 32. The mean of two of the numbers is 28. Discuss the steps you would follow to find the mean of the other two numbers.” Students engage in non-routine application of 6.SP.3 (Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number).

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations in 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.

All three aspects of rigor are present independently throughout the materials. For example:

• In Section 3.4, Multiplying Decimals Fluently, Math Sharing, page 132, students use a concrete approach to multiply decimals. The materials state, “The model on the right shows 0.2 × 0.6 = 0.12. 1. (Find two other decimals that give a product of 0.12. 2.) Find two decimals that give a product of 0.36. Share with your classmates how you found the decimals.” Students develop conceptual understanding of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation).

• In Section 5.2, Real-World Problems: Rates and Unit Rates, Activity, page 252, students compare unit rates in a real-world situation. The materials state, “1) Press your fingers firmly on your wrist to feel your pulse. Count the number of times your heart beats in 15 seconds while your partner uses a stopwatch to take the timing. 2) Find your resting heart rate per minute. Compare your unit heart rate to your partner’s unit heart rate. What do you notice? 3) Find your heart rate per minute after doing 30 jumping jacks. Compare your unit heart rate to your partner’s unit heart rate. What do you notice?” Students engage in application of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems).

• In Section 6.4, Real-World Problems: Percent, Chapter Review, page 343, students practice converting between fractions, decimals, and percents. Problem 1 states, “Express each percent as a fraction in simplest form. 46%” Problem 4 states, “Express each percent as decimal 34%.” Problem 7 states, “Express each fraction as a percent $$\frac{17}{20}$$.” Problem 10 states, “Express each decimal as a percent 0.02.” Students engage in procedural skill and fluency of 6.RP.3c (Find a percent of a quantity as a rate per 100).

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

• In Section 3.2, Real-World Problems: Fractions, Learn,Problem 1, page 103, students divide a whole number by a fraction to solve real-world problems. The problem states, “A chef cooks 12 pounds of pasta each day. She uses $$\frac{3}{16}$$ pounds of pasta for each serving she prepares. How many servings of pasta does she prepare each day?” Students build procedural skill and fluency and apply the mathematics of 6.NS.1 (Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions).

• In Section 6.3, Percent of a Quantity, Try, Problem 1, page 326, students practice finding the whole given a quantity and its percent. The problem states, “27% of the students in a school are in Grade 6. There are 540 Grade 6 students. How many students are there in the school?” A tape diagram to assist with filling in missing values is provided. Students develop conceptual understanding and build procedural skill and fluency of 6.RP.3c (Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent). 

• In Section 8.4, Solutions of Simple Inequalities, Independent Practice, Problem 27, page 92, students use inequalities to represent real-world situations. The problem states, “In the inequality q 24.3, q, represents the possible weights, in pounds, of a package. a) Is 24.4 a possible value of q? Explain. b) Is $$20\frac{7}{10}$$ a possible value of q? Explain. c) Use a number line to represent the solution set of the inequality. State the greatest possible weight of the package.” Students develop conceptual understanding and apply the mathematics of 6.EE.5 (Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true using substitution to determine whether a given number in a specified set makes an equation or inequality true).

The materials reviewed for Math in Focus: Singapore Math Course 1 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 the opportunity to construct viable arguments in connection to grade-level content leading to the intentional development of MP3, identified as mathematical habits in the materials. However, the teacher guidance is often repetitive, not specific, and often distracts from the intentional development of MP3. Examples include:

• In Lesson 2.2: Number Lines and Negative Numbers, Independent Practice, Problem 38, page 68, students construct viable arguments to understand negative numbers. The problem states, “Your friend says that the statement 0 < -15 is correct. Explain why this statement is correct.” Teacher guidance states, “Assess students’ understanding of negative numbers and ability to compare numbers using > or <.” While this teacher guidance does not intentionally develop MP3, students do have the opportunity to construct a viable argument.

• In Section 13.6, Real-World Problems: Measures of Central Tendency and Variability, Independent Practice, Problem 2, page 371, students construct viable arguments about which measure of central tendency to use to solve real-world problems. The problem states, “The table shows the results of a survey carried out on 80 families. a) Find the mean, median, and mode. b) Which measure of central tendency best describes the data set? Explain your answer.” A table showing the number of children and number of families is provided. Teacher guidance states, “Assess students’ ability to decide which measure of central tendency to use to solve real- world problems.” While this teacher guidance does not intentionally develop MP3, students do have the opportunity to construct a viable argument.

Students have the opportunity to critique the reasoning of others in connection to grade-level content leading to the intentional development of MP3, identified as mathematical habits in the materials. Examples include:

• In Section 3.1, Dividing Fractions, Math Talk, Page 99, students critique the reasoning of others when dividing fractions. The materials state, “Connor’s solution to $$\frac{6}{7} ÷ \frac{3}{8}$$ is as shown. Do you agree with Connor’s solution? Why? Explain how you would divide a fraction by a fraction.” Teacher Guidance states, “Fraction models help students to visualize dividing a fraction by a fraction, but ultimately they should apply the rule of using the reciprocal of the divisor, instead of the dividend. In pairs, have students discuss if they agree with Connor’s solution. Invite students to share their strategies to divide a fraction by a fraction without using fraction circles. Be sure students know the mathematics behind dividing by a fraction and cross-simplification.”

• In Section 4.1, Comparing Two Quantities, Try, Problem 2, page 170, students critique the reasoning of others when writing ratios to compare two quantities with the same unit. The problem states, “Ms. Clark buys 4 bags of apples and 7 bags of oranges. Each bag has an equal number of fruits. The ratio of the number of apples to the number of oranges is ____:____. The number of apples is ____ the number of oranges.” Math Talk note states, “Zoe says that for every 7 oranges, there are 4 apples. Is she correct? Why?” Teacher guidance states, “Is Zoe correct? Why? Reinforce students’ understanding of ratios by using a visual to illustrate the situation.”

Math Journal Activities provide opportunities for students to engage in the intentional development of MP3. Examples include:

• In Chapter 6, Percent, Math Journal, page 341, students construct viable arguments about writing percents. The problem states, “Sara and Kyle checked some books out of the library. 20% of the books Sara checked out were fiction books, and 40% of the books Kyle checked out were fiction books. Your friend thinks that Kyle checked out more fiction books than Sara. Explain the error in your friend’s thinking. Use an example to support your reasoning.” Teacher guidance states, “Review with students the key concepts learned in this chapter. Encourage students to write down an example to support their reasoning. You may want to pose these questions to students who are struggling to reason out their answers: Can your friend be correct? Can he or she be wrong too? Can you think of more examples to support your reasoning?” 

• In Chapter 7, Algebraic Expressions, Math Journal, page 41, students construct viable arguments as they simplify expressions. The materials state, “William simplified the two expressions shown. 10w - 5w + 2w = 10w - 7w = 3w 12 + 24x = 12(2x). Are William’s answers correct? If not, explain why they are incorrect?” Teacher guidance states, “Review with students the various strategies learned in this chapter. Encourage students to use multiple strategies (visuals and/or substitution) to determine the correctness of the situation. Walk among the students to provide support as needed. You may want to post these questions to students who are struggling to reason out their answers: ‘How can we determine if William is correct? What can we say to William to help him identify his error?’ You may also want to provide visuals or encourage students to use models to support their reasoning.”

Materials identify focus Mathematical Habits in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and in the Section Objectives. However, these mathematical habits are not intentionally addressed in the activities and problems. Examples include:

• Section 5.4, Average Speed, is noted as addressing MP3 on pages 271-176 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no opportunities for students to construct arguments or critique the reasoning of others is provided in the lesson.

• Section 6.2, Fractions, Decimals, and Percents, is noted as addressing MP3 on pages 315-322 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no opportunities for students to construct arguments or critique the reasoning of others is provided in the lesson.