4th Grade - Gateway 2

The materials reviewed for Math in Focus: Singapore Math Grade 4 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The materials develop procedural skill and fluency throughout the grade level. Examples include:

• In Section 1.4, Adding and Subtracting Multi-Digit Numbers, Engage, Problems a and b, page 41, students show and explain their strategies when describing a number as the sum of their place value. The materials state, “a) Use (place-value chips) to show the 74,528. Then, fill in each blank. 74,528 = 6 ten thousands ___ thousands 5 hundreds ___ ones. b). Fill in each blank. 68,364 = ___ten thousands 18 thousands ___ hundreds 26 tens 4 ones.” Students begin to develop procedural skill and fluency of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm).

• In Section 1.6, Real World Problems: Addition and Subtraction, Learn, Problem 2, page 63, students use bar models to show and explain their strategies for adding multi-digit whole numbers. The problem states, “Store A sold 16,245 baseball cards last year. Store A sold 5,648 more baseball cards than Store B. How many baseball cards did they sell in all?” Students begin to develop procedural skill and fluency of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm).

• In Section 2.5, Factors, Hands-on Activity, Problems 5 and 6, page 169, students work with factors and multiples. Problem 5 states, "Take 16 (counters) and arrange them in rows and columns.” Problem 6 states, “Use multiplication to list the different ways in the space below. Then, list the factors of 16. The factors are ___, ___, ___, ___, and ___.” Students begin to develop procedural skill and fluency of 4.OA.4 (Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors).

• In Section 3.2, Comparing and Ordering Fractions, Hands-on Activity, Activity 1, Problem 1, page 249, students compare numbers using bar models and benchmarks. Using models to order fractions, “Shade the correct number of parts for each fraction. Fractions represented are $$\frac{1}{4}, \frac{1}{12}, \frac{5}{6}$$. 2). Order the fractions from greatest to least: $$\frac{\Box}{greatest}$$, ____,$$\frac{\Box}{least}$$." In Independent Practice, Problem 1, page 251, states, "Use equivalent fractions to compare each pair of fractions. Which is less, $$\frac{3}{8}$$ or $$\frac{1}{2}$$?” Students develop procedural skill and fluency of 4.NF.2 (Compare two fractions with different numerators and denominators).

Materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   

• In Section 1.4, Adding and Subtracting Multi-Digit Numbers, Independent Practice, Problem 7, page 47, states, “Solve. Show your work 21,574 and 183,261.” Students independently demonstrate the procedural skill of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm).

• In Section 1.6, Real-World Problems: Addition and Subtraction, Independent Practice, Problem 3, page 75, states, “Last year, School A raised $24,950 for charity. School B raised $8,504 more than School A. School C raised $12,080 less than School B. How much did Schools B and C raise in all last year?” Students independently demonstrate the procedural skill of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm).

• In Section 4.5, Fractions and Decimals, Independent Practice, Problem 17, page 392, states, “Express each decimal as a fraction. Then, add. Express the answer as a decimal, 0.59 + 0.6.” Students demonstrate procedural skill and fluency of 4.NF.5 (Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fraction with respective denominators 10 and 100) and 4.NF.6 (Use decimal notation for fractions with denominators 10 or 100).

• In Section 9.1, Making and Interpreting a Table, Independent Practice, Problem 2, page 322, states, “Use the data in the table to answer each question. Each student in a class chose his or her favorite type of pie. The data is represented in the table below. c. How many more students prefer apple pies to lemon pies? d. How many students chose apple pie and strawberry pie as their favorites?” Students independently demonstrate procedural skill and fluency of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm). 

Grade 4 materials contain a separate Fact Fluency book so students can independently practice specific strategies to promote procedural skill and fluency. Examples include:

• In Chapter 1, Fact Fluency, Working with Whole Numbers, Problems 1-5, page 5, students add and subtract multi-digit numbers with regrouping. The materials state, “Match each problem to its answer. 1) 14,100 - 18, 2) 13,999 + 3, 3) 14,764 - 735, 4) 13,900 + 192, 5) 15,082 - 1,027.” Answer choices: “14,002, 14,029, 14,092, 14,082, and 14,055.” Students independently practice procedural skill and  fluency of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm).

• In Chapter 5, Fact Fluency, Conversion of Measurements, 5c Conversion of Length, Problems 1-3, page 65, students compute conversions of length. The materials state, “Fill in the blanks, ____ feet = 3 yards. Problem 2, ____feet = 2 miles. Problem 3, ____inches = 2 feet.” Students independently practice procedural skill and fluency of 4.MD.1 (Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; 1b, oz. l, ml; hr, min, sec).

• In Chapter 6, Fact Fluency, Area and Perimeter, page 77, students calculate the area perimeter of a square and a rectangle given the length and width. The materials state, “Sketch and label the square or rectangle. Then, complete the table. The table has the following lengths/widths: “L: 5in, W: 4in; L: 4ft, W: 3ft; L: 8ft, W: 2ft; L: 9cm, W: 2cm; L: 6mm, W: 6mm.” Students independently practice procedural skill and fluency of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real-world and mathematical problems).

• In Chapter 9, Fact Fluency, Tables, Fractions, and Decimals, 9e Real World Multiplication and Division Problems, page 102, students solve real-world multiplication and division problems. The materials state, “The Grade 4 students at an elementary school sold rolls of wrapping paper to raise money for a field trip. [A box showing Clue 1, Clue 2 and Clue 3 is shown.] Fill in the Blanks. Use the clues to help you. 1) Mr. Martinez’s class sold ____ rolls of wrapping paper. 2) Mr. Jones’s class sold ____ rolls of wrapping paper. 3) Mr. Thomas’ class sold ____ rolls of wrapping paper. 4) Each roll of wrapping paper costs $6. How much money did each class raise? Mr. Martinez’s class raised $____, Mr. Jones’s class raised $____, Mr. Thomas’ class raised $____,. 5) It costs $300 for the entire Grade 4 to go on the field trip. Did they raise enough money? Explain your thinking.” Students independently practice procedural skill and fluency of 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison).

The materials reviewed for Math in Focus: Singapore Math Grade 4 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Students have opportunities to independently demonstrate routine application of the mathematics. Examples include:

• In Section 1.6, Real-World Problems: Addition and Subtraction, Engage, page 61, students use a bar model to express a real-world problem. The materials state, “School X has 4,400 students enrolled. School Y has 250 more students enrolled than School X. School Z has 250 fewer students enrolled than School X. Draw a bar model to show the number of students in each school.” Students engage in the routine application of 4.OA.3 (Solve multistep word problems posed with whole numbers and have whole-number answers using the four operations, including problems in which remainders must be interpreted).

• In Section 2.7, Multiplication and Division, Try, Problem 1, page 197, students solve real-world problems using the bar model to represent the problem and identify the operation needed to solve it. The problem states, “Irene had $3756 to spend on office furniture. She bought a sofa for $1,195 and 6 chairs for $128 each. a) How much money did she spend altogether? b) How much money did she have left.” Students are provided with bar models for scaffolding. Students engage in the routine application of 4.OA.3 (Solve multistep word problems with whole numbers and have whole number answers using the four operations, including problems in which remainders must be interpreted).

• In Section 5.3, Real-World Problems: Customary Units of Measure, Independent Practice, Problem 4, page 48, students solve real-world problems involving weight conversions. The problem states, “A family of elephants lives in a zoo. The female adult elephant gives birth to 2 baby elephants weighing 220 pounds each. The adult male elephant weighs 6 tons and the adult female weighs 4 tons. Find the total weight of the family of elephants in pounds.” Students independently engage in the routine application of 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time liquid volumes, masses of objects, and money including problems involving simple fractions or decimals, and problems that require expressing measurements given a larger unit in terms of a smaller unit).

• In Section 5.7, Time, Engage, page 74, students solve problems involving elapsed time. The materials state, “The clock shows half past 12 in the afternoon. What is the time shown on the clock 80 minutes later? What is another way to show the time? Share your ideas with your partner.” Students engage in the routine application of 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit).

• In Chapter 7, Angles and Line Segments, Performance Task, Problem 3, page 251, students add missing pathways that include parallel and perpendicular line segments. The problem states, “A designer drew a model of some of the pathways in an apartment complex. Use your understanding of perpendicular and parallel lines to add the pathways that are missing in the drawing. [A drawing of a trapezoid is shown, with various points and lines.] a)  Draw a pathway that is parallel to PR through point Q. b) Draw a pathway that is parallel.” Students engage in the routine application of 4.G.1 (Draw points, line segments, rays, angles, and perpendicular and parallel lines). 

Students have opportunities to independently demonstrate non-routine application of the mathematics.  Examples include:

• In Section 2.4, Real-World Problems: Multiplication and Division, Hands-on Activity, Problems 1 and 2, page 155, students work in pairs to solve a real-world problem involving multiplication. Problem 1 states, “Write a real-world problem for the bar model shown below.” Problem 2 states, “Ask your partner to solve the problem and explain how they check their answers.” A bar model showing 2,568 divided by 7 is shown. Students engage in the non-routine application of 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison) and 4.NBT.6 (Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors).

• In Chapter 3, Put on Your Thinking Cap, Problem 2, page 323, students use number sense to find the greatest difference between two mixed numbers, without repeating digits,  The problem states, “Fill in each box with the digits 2, 3, 4, 6, 8, or 9 to form two mixed numbers that have the greatest possible difference. Each digit can be used only once. Then find the difference. Express the answer in simplest form. ___ ___ $$\frac{\Box}{10}$$ - ___ $$\frac{\Box}{10}$$ = ___.” Students independently engage in the non-routine application of 4.NF.3c (Add and subtract mixed numbers with like denominators). 

• In Section 6.1, Area and Unknown Sides, Engage, page 117, students solve mathematical problems involving area and perimeter. The materials state, “Draw four different rectangles on a square grid. Use a table to make a list of length and width of each rectangle. Find the area of each rectangle. Discuss the relationship between the length and the area with your partner. What pattern do you notice?” Students engage in the non-routine application of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real world and mathematical problems).

• In Section 6.2, Composite Figures, Engage, page 129, students use geoboards to create two rectangles with different lengths but the same perimeters. The materials state, “Rectangle A is 4cm longer than Rectangle B but their perimeters are the same. How is this possible? Explore all possibilities using a geoboard. Explain how you arrive at your answer.” Students engage in the non-routine application of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real-world and mathematical problems).

The materials reviewed for Math in Focus: Singapore Math Grade 4 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. 

All three aspects of rigor are present independently throughout Grade 4. Examples where instructional materials attend to application, procedural skill and fluency, or conceptual understanding, include: 

• In Section 4.3, Comparing and Ordering Decimals, Hands-on Activity, Activity 1, Problems 1-3, page 367, students develop conceptual understanding of place value to compare decimals to hundredths. Problem 1 states, “Use (place value disk pictures) 1, 0.1, 0.01 to show 8.5 and 9.2. Then, mark 8.5 and 9.2 on the number line.” Problem 2 states, “Ask your partner to use ‘greater than’ or ‘less than’ to describe the decimals.” Problem 3 states, “Trade places. Repeat 1 and 2 with 13.28 and 13.47.” Number lines are provided for Problems 1 and 3. Students develop conceptual understanding of 4.NF.7 (Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole).

• In Section 5.3, Real-World Problems: Customary Units of Measure, Independent Practice, Problem 2, page 47, students apply the four operations to solve real-world problems. The problem states, “A pound of ground turkey cost $6. A pound of ground chicken cost $4. A chef brought 8 pounds of ground turkey and 4 pounds of ground chicken to mix and make hamburger patties. How much did the ground turkey and ground chicken cost altogether?” Students use the four operations to solve word problems, masses of objects and problems that require expressing measurements given in a larger unit in terms of a smaller unit. (4.MD.2)

• In Section 7.2, Drawing Angles to 180°, Hands-On Activity, Problem 1, page 198, students develop procedural skill and fluency as they use a ruler and a protractor to draw angles.  The problem states, “Use a ruler and protractor to draw each of the following angles on the next page. a) 50$$\degree$$, b) 35$$\degree$$, c) 90$$\degree$$, d) 140$$\degree$$.” Students develop procedural skill and fluency with 4.MD.6 (Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure). 

Multiple aspects of rigor are engaged simultaneously to develop a student's mathematical understanding. Examples include: 

• In Section 1.1, Numbers to 100,000, Engage, page 11, students develop conceptual understanding alongside procedural skill and fluency as they use concrete materials to express a 4-digit number in word and expanded form. The materials state, “Use (base-ten blocks) to show 3,245. Write the number in expanded form and word form.” Students read and write multi-digit whole numbers using number names and expanded form. (4.NBT.2)

• In Chapter 3, Extra Practice and Homework, Fractions and Mixed Numbers, Activity 9, Problem 3, page 117, students develop conceptual understanding alongside application as they solve real-world problems involving fractions. The problem states, “Solve. Show your work. Draw a bar model to help you. Express each answer in simplest form. Farrah and Andrea baked 1 kilogram of bread in all. Farrah baked $$\frac{3}{10}$$ kilogram of bread. How much more bread did Andrea bake than Farrah?” Students solve word problems involving addition and subtraction of fractions referring to the same whole and having the same denominator, by using visual fraction models and equations to represent the problem. (4.NF.3d)

• In Section 6.1, Area and Perimeter, Try, Problem 3, page 111, students develop procedural skill and fluency alongside application to find the perimeter of a rectangle within a real-world context. The problem states, “A rectangular pool has a length of 15 meters and a width of 8 meters. What is the perimeter of the pool? Students apply the area and perimeter formulas for rectangles in real-world and mathematical problems. (4.MD.3)

The materials reviewed for Math in Focus: Singapore Math Grade 4 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 construct viable arguments and 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:

• In Chapter 2, Multiplication and Division, Math Journal, page 207, students critique the reasoning of others when they identify and correct the mistake in another student’s strategy. The materials state, “Ms. Scott had 708 pennies. She had twice as many pennies as Mr. Perez. Ms. Young had 3 times as many pennies as Mr. Perez. How many pennies did Ms. Young have? Eric’s solution: 708 × 2 = 1,416; 1,416 × 3 = 4,248. Ms. Young had 4,248 pennies. a. Explain the mistake in Eric’s solution. b. Show how you would solve the problem.” Teacher notes include, “Students have an opportunity to perform error analysis to correct Eric’s thinking, and solve the problem for themselves.” 

• In Section 3.1, Equivalent Fractions, Hands-on Activity, Page 233, Problem 2, students construct viable arguments to explain why fractions are equal. The problem states, “Explain why the fractions are equal. Show how to use one fraction to find the other.” Students are provided with two images each representing a fraction, $$\frac{1}{2}$$ and $$\frac{2}{4}$$. Teacher guidance includes, “In 2, students will defend and justify their thinking, as well as critique the reasoning of their classmates.” and “You may want to conclude the activity with the following question. How would you explain what you are doing to someone who is absent today?” 

• In Section 3.2, Comparing and Ordering Fractions, Hands-on Activity, Problem 2, page 241, students construct viable arguments as they work in pairs to compare fractions and consider which method they prefer. The problem states, “For each pair of fractions, explain to your partner why you chose that method. Show how you used the method to compare the fractions. a) $$\frac{7}{8}$$ and $$\frac{1}{6}$$ , b) $$\frac{11}{12}$$ and $$\frac{2}{3}$$.” Teacher guiding questions include, “Which method will you choose for the first pair of fractions, and why? Which method did you choose for the second pair of fractions, and why? When do you choose one method over another?” 

• In Chapter 3, Fractions and Mixed Numbers, Math Journal, Page 321, students critique the reasoning of others. The materials state, “A recipe for an apple pie uses $$\frac{3}{4}$$ cup of flour. Jackson wants to make 2 apple pies. He only has a $$\frac{1}{4}$$-cup measuring cup. He says he has to use the cup 6 times to get the amount of flour he needs. Show why his reasoning is correct.” Teacher notes include, “Students have an opportunity to prove Jackson’s thinking. Have them draw bar models in addition to recording the math sentence used to achieve the answer. You may want to pose these questions to students who are struggling with constructing sable arguments. What is the problem asking you to do?” and “What do you know that can help you solve this problem?”

• In Chapter 4, Decimals, Math Journal, page 393, students construct viable arguments as they compare decimals. The materials state, “Chris and Mary compare 0.23 and 0.3. Chris says 23 is greater than 3. So, 0.23 is greater than 0.3. Mary says 23 tenths is greater than 3 tenths. So, 0.23 is greater than 0.3. Do you agree? Why or why not? Explain.” Teacher guidance includes, “Is there a tool you can use to check Chris and Mary’s thinking? If so what is it? If they have trouble beginning remind them of the four-step problem-solving model used to work through the problems. You may want to encourage discussion by posing this question to students. What do you know?” 

• In Chapter 5, Conversion of Measurements, Math Journal, Page 85, students construct viable arguments as they determine if equations associated with measurement conversions are correct. Teacher guidance includes, “review the various strategies to convert both metric and customary units of measure of length, mass, and volume.” The teacher “encourages students to work independently” and poses questions for students who are struggling to construct viable arguments. Teachers ask, “What is the relationship between the units of measure? How does this help us know which operation to use to convert between them? How do you know that you are correct? How can you prove your thinking? Look at the following conversions. Which is correct? Explain. A) 2 gal = 10 pt , B) 3 lb = 48 oz, C) 5 yd = 8 ft, D) 12 km = 12 m, E) 3 L = 3,000 mL.” 

Materials identify focus Mathematical Habits in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and in the Section Objectives. However, these mathematical habits identified as evidence of MP3 are not intentionally addressed in the activities and problems. Examples includes:

• Section 3.1, Equivalent Fractions, is noted as addressing MP3 on page 231 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 7.5, Drawing Perpendicular and Parallel Line Segments, is noted as addressing MP3 on page 223-238 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 8.3, Symmetric Shapes and Lines of Symmetry, is noted as addressing MP3 on page 273-284 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.