4th Grade - Gateway 2

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts and provide opportunities for students to independently demonstrate conceptual understanding throughout Grade 4. 

Materials develop conceptual understanding throughout the grade level. According to Course Summary, Learn More About Fishtank Math, Our Approach, “Procedural Fluency AND Conceptual Understanding: We believe that knowing ‘how’ to solve a problem is not enough; students must also know ‘why’ mathematical procedures and concepts exist.” Each lesson begins with Anchor Tasks and Guiding Questions, designed to highlight key learning aligned to the objective and to support the development of conceptual understanding through student discourse and reflection. This is followed by a Problem Set, Homework and Target Task. Examples include: 

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 7, Anchor Task, Problem 2 states,  “Write each of the following in expanded and written form. 1. 27,085; 2. 601,408; 3. 7,056.” Guiding Questions include, “27,085 has a 0 in the hundreds place. How did you account for that when you wrote the number in expanded form? In written form? How can the placement of the comma in the number help with determining how to write the number in word form? How can reading the number out loud or to a partner help with determining how to write the number in word form?” This problem and Guiding Questions provide the teacher an opportunity to help students develop conceptual understanding of 4.NBT.2 (Read and write multi-digit whole numbers in base ten numbers, number names, and expanded form).

• In Unit 2, Multi-Digit Multiplication, Lesson 5, Anchor Task, Problem 2, students multiply a whole number of up to four digits by a one-digit whole number. It states, “Solve. a. 5 × 4; b. 5 × 40; c. 5 × 400; d. 5 × 4,000. 2. What do you notice about #1? What do you wonder?” Guiding Questions include, “How can you use the patterns you noticed in Anchor Task #1 to solve? How is this problem different from that in Anchor Task #1? I think 5 × 4,000 = 2,000. What mistake did I make? How can you use “times as many” to describe these equations?” This problem shows opportunities for students to engage with teacher support and/or guidance while developing conceptual understanding of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models).

• In Unit 6, Decimal Fractions, Lesson 8, Anchor Tasks, Problem 2, Fill in the blank with <, =, > to complete the equation. Justify your comparison. a. 1.24____2.24; b. 2.38____2.83; c. 4.38____4.5; d. 6.37____6.3; 3. 10.0____10.00; f. 15.2____15.02.” Guiding Questions include, “How can you justify your answer with a picture? How can you justify your answer using the meaning of a decimal as a fraction? Is it easier to represent particular problems with certain models?”  This activity helps 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).

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Problem Sets and Homework Problems can be completed independently during a lesson. Target Tasks, or end of lesson checks for understanding of key concepts, are designed for independent completion. Many of these problems provide opportunities for students to independently demonstrate conceptual understanding. Examples include:

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 1, Problem Set, Problem 2, students recognize a digit in one place represents ten times what it represents in the place to its right. It states, “Find 412 in your 1,000 book. a. Which hundreds chart is it on? b. Which two tens is it between? c. What would you have to change to make its neighbor to its left?” This problem shows conceptual understanding of 4.NBT.1 (Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right).

• In Unit 4, Fraction Equivalence and Ordering, Lesson 11, Problem Set, Problem 6, students compare and analyze fractions with different denominators as they solve, “Andrea bought a bucket of colored chalk.The list below shows the fraction of each color of chalk in the bucket. $$\frac{2}{6}$$ are yellow; \frac{5}{12} are blue; \frac{3}{12} are green. a. Which is greater, the amount of yellow chalk in the bucket or the amount of green chalk in the bucket? b. Andrea told Michelle that less than \frac{1}{2} the chalk in the bucket is blue. Michelle said she is mistaken. Who is correct? Explain why you chose your answer.” This problem shows conceptual understanding of 4.NF.2 (Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model).

• In Unit 5, Fraction Operations, Lesson 4, Target Task, Problem 2, students “Solve. Show or explain your work. Of the computer games Lynne owns, \frac{4}{12} are sports games and \frac{3}{12} are educational. What fraction of the games are neither sport games nor educational games?” This problem will show students’ conceptual understanding of 4.NF.3.d (Solve word problems involving addition and subtraction of fractions).

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for developing procedural skills and fluency while providing opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. 

According to Teacher Tools, Math Teacher Tools, Procedural Skill and Fluency, “In our curriculum, lessons explicitly indicate when fluency or culminating standards are addressed. Anchor Problems and Tasks are designed to address both conceptual foundations of the skills as well as procedural execution. Problem Set sections for relevant standards include problems and resources that engage students in procedural practice and fluency development, as well as independent demonstration of fluency. Skills aligned to fluency standards also appear in other units after they are introduced in order to provide opportunities for continued practice, development, and demonstration.”

Opportunities to develop procedural skill and fluency with teacher support and/or guidance occurs in the Anchor Tasks, at the beginning of each lesson, and the Problem Sets, during a lesson. Separate fluency activities can be used independently or with teacher support.  Examples Include:

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 15, Fluency Activities,  students play the game, “Get to 1000,” with a partner. It states, “Students shuffle cards and put them face down. Each student selects 6 cards and makes two three-digit numbers that they then add together. The cards are added to the bottom of the deck after each turn. After the first round, students pull another 3 cards to make a three-digit number to add to their running total. The objective is to be the student to first reach exactly 1,000. In order to reach exactly 1,000, a student may choose to take just 1 or 2 cards instead of 3. If the only values a student can make from the cards they’ve chosen is larger than what is needed to reach 1,000, they lose their turn.” This activity helps students develop 4.NBT.4 (Fluently add and subtract multi-digit numbers using the standard algorithm).

• In Unit 4, Fraction Equivalence and Ordering, Lesson 1, Anchor Tasks, Problem 1 states, “Look at the patterns that a skip-counting sequence made: (hundred chart.) What will the next three numbers in the sequence be? Will 62 be in the pattern? How do you know? How many fours are equal to the number 32?” Teacher Guiding Questions include, “What did you notice about the numbers in the sequence in the hundred chart? Can you write a multiplication sentence to match #3? All of the shaded numbers are multiples of 4. What patterns do you notice about all of the multiples of 4? What are the multiples of 9? What patterns do you notice about them?” This problem provides an opportunity for students to develop fluency of 4.OA.4 (Find all factor pairs for a whole number in the range 1—100).

• In Unit 5, Fraction Operations, Lesson 3, Anchor Task, Problem 2 states, “First, estimate the following sums. Then add. a. \frac{2}{8}+\frac{3}{8} b. \frac{1}{6}+\frac{3}{6}”. The Guiding Questions prompt teachers to ask, “Before adding, do you expect our sum to be greater than or less than one half? Why? How can we think of this in unit form? How will that help us solve? What picture can we draw to model the problem? Can we rewrite our solution using larger units? Based on our estimate, is our computed sum reasonable? Why or why not? (Ask similar questions for part (b).)” This problem provides an opportunity for students to develop procedural skill and fluency of 4.NF.3 (Understand addition and subtraction of fractions as joining and separating parts referring to the same whole).

The instructional materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Problem Sets and Homework can be completed independently during a lesson. Target Tasks, or end of lesson checks for understanding, are designed for independent completion. Fluency specific activities also provide opportunities for independent application of fluency. Examples include:

• In Unit 3, Multi-Digit Division, Lesson 4, Target Task, students divide multi-digit numbers by one digit divisors. The task states, “Solve. Show or explain your work. 1. 87 ÷ 4; 2. 294 ÷ 6; 4,256 ÷ 7.” This problem provides an opportunity for students to independently demonstrate procedural skills and fluency of 4.NBT.6 (Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors).

• In Unit 4, Fraction Equivalence and Ordering, Lesson 1, Target Task, Problem 1, students find multiples. The problem states, “Select the list of numbers that are all multiples of 9. A. 9, 27, 35, 63; B. 9, 48, 81, 90; C. 18, 36, 45, 64; D. 18, 54, 72, 99.” This problem provides an opportunity for students to independently demonstrate fluency of 4.OA.4 (Find all factor pairs for a whole number in the range 1—100). 

• In Unit 5, Fraction Operations, Lesson 12, Target Task states, “Solve. Show or explain your work. 1. 3\frac{2}{6}+\frac{3}{6}; 2. 6\frac{9}{10}+\frac{3}{10}.” These problems provide an opportunity for students to independently demonstrate procedural skills of 4.NF.3 (Add and subtract mixed numbers with like denominators).

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

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Anchor Tasks, at the beginning of each lesson, routinely include engaging single and multi-step application problems. Examples include:

• In Unit 2, Multi-Digit Multiplication, Lesson 3, Anchor Tasks, Problem 2, students engage in solving non-routine application problems as they interpret a multiplication equation as a comparison (4.OA.1) and multiply or divide to solve word problems involving multiplicative comparisons (4.OA.2). The problem states, “Jade has $63. Keith has $9. How many times as much money does Jade have as Keith? Represent the situation as an equation to solve. Guiding Questions for teachers include, “How can we represent this situation with a tape diagram? Where are the various parts of the problem represented in your tape diagram? How can we represent this situation with an equation? How can you use a letter to represent the unknown? Where do you see ‘seven times as many’ in the equation? May gets $4 in allowance every week. Her older sister, Tatiana, gets $12 in allowance every week. How many times more does Tatiana get in allowance than May? How can we represent this situation with a tape diagram? With an equation?” 

• In Unit 2, Multi-Digit Multiplication, Lesson 7, Anchor Task, Problem 1, students solve routine problems involving multiplication of 2-digit numbers by 1-digit numbers (4.NBT.5). The problem states, “Mr. Wynn gets some butcher paper from the teacher supply room to put down the length of the hallway. The butcher paper is 3 feet tall and 20 feet long. When he puts it up in the hallway, he realizes he needs a little more to fully cover the wall. The extra piece he needed is 3 feet tall and 3 feet long. a. How many square feet of butcher block paper did Mr. Wynn put in the hallway? b. What was the total length of width of the space Mr. Wynn ended up covering?” Guiding Questions for teachers include, “What model can we draw to represent the problem? How can we record that work with equations? Does finding the area of each separate piece of butcher block paper and adding those areas together give the same result as finding the overall length and width of the space Mr. Wynn needed to cover and finding the overall area? Why?”

• In Unit 3, Multi-Digit Division, Lesson 12, Anchor Tasks, Problem 4 provides students the opportunity to solve routine problems involving division in which remainders must be interpreted (4.OA.3). The problem states, “Joao loves The Diary of a Wimpy Kid and convinces Ms. Glynn to buy as many as she can for his classmates to read. Ms. Glynn has $150 left in her classroom budget to buy them. Each book costs $8. How many copies can she get?” Guiding Questions for teachers include, “What is happening in this situation? What can you draw to represent this situation? What quantities and relationships do we know? What is the question asking you to find out? What equation can we use to represent each part of the problem? What letter should we use to represent the unknown? How did you find the answer to the question? Did anyone find the answer differently? How did you interpret the remainder in the context of this problem? Why did you discard the remainder? Is your answer reasonable? How do you know?”

• In Unit 7, Unit Conversions, Lesson 12, Anchor Task, Problem 1, students engage in solving non-routine application problems as they use the four operations to solve word problems involving distance, intervals of time, liquid volumes, masses of objects, and money (4.MD.2). The problem states, “Watch the following video: The Water Boy (Act-1). Here is how much water the water bottle holds: [picture shows 1.5 liter water bottle] Here is how much water is left in the bottle: [picture shows a measuring cup with water just below the 900 ml mark]. "How much water did he drink?” Guiding Questions for teachers include, “How can you determine how much water he drank? Is there more than one way to solve?”  

Materials provide opportunities within Problem Sets and Homework, and Daily Word Problems for students to independently demonstrate multiple routine and non-routine applications throughout the grade level. Target Tasks, or end of lesson checks for understanding, are designed for independent completion. Examples include:

• In Unit 1 Place Value, Rounding, Addition, and Subtraction, Lesson 18, Target Task, students independently solve non-routine multistep word problems (4.OA.A). The problem states, “Quarterback Brett Favre passed for 71,838 yards between the years 1991 and 2011. His all-time high was 4,413 passing yards in one year. In his second-highest year, he threw 4,212 passing yards. How many passing yards did Brett Favre throw in the remaining years? Is your answer reasonable? Explain.”

• In Unit 5, Fraction Operations, Lesson 16, Additional Practice, Word Problem Practice, students independently solve routine fraction word problems with addition and subtraction (4.NF.3d). “Negin has watched 415 hours less TV than Ji-Yu. Negin has watched 1315 hours of TV. How many hours of TV did Ji-Yu watch?”

• In Unit 5, Fraction Operations, Lesson 20, Homework, Problem 6, students independently engage in solving non-routine application involving problems involving addition and subtractions of fractions (4.NF.3), and multiplication of a fraction by a whole number (4.NF.4). The problem states, “a. Kayla, Jim, and Maria each ran after school last week. Kayla ran \frac{2}{3} mile each day after school for 5 days. How many total miles did Kayla run last week? b. Last week Jim trained to run long distance. Each day, he ran \frac{3}{4} mile before and \frac{3}{4} mile after school for 5 days. How many total miles did Jim run last week? c. Maria ran \frac{3}{4} mile each day for 3 days and \frac{1}{4} mile each day for 2 days. How many total miles did Maria run last week?”

• In Unit 6, Decimal Fractions, Lesson 14, Target Task, students independently solve routine word problems involving money (4.MD.2). The problem states, “Solve. Show or explain your work. Write your answer in decimal form. David needs $4 to buy some candy he wants after school. He searched the couch cushions and found 6 quarters, 4 dimes, and 26 pennies. How much more money does he need to make?”

The materials reviewed for Fishtank Plus Math Grade 4 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 Grade 4. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 14, Anchor Task, Problem 2, students develop procedural skill and fluency as they add multi-digit numbers using rounding strategies. The problem states, “Estimate. Then solve. a. 207,426 + 128,744. b. 252, 393 = ◻ – 747,607. c. k  = 94,989 + 619,732 + 4,506.” (4.NBT.4: Fluently add and subtract multi-digit whole numbers using the standard algorithm.)

• In Unit 3, Multi-Digit Division, Lesson 8, Problem Set, Problem 2, students develop conceptual understanding as they solve three-digit dividend division problems. The problem states, “Complete the steps to find the quotient of 492 ÷ 6. Step 1. ( ___ ÷ 6) + (180 ÷ 6) + (12 ÷ 6). Step 2. ___ + ___ + 2. Quotient ___. ” (4.NBT.6: Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.)

• In Unit 7, Unit Conversion, Lesson 4, Target Task, Problem 2, students convert metric units (length, mass, and capacity) as they solve application problems. The problem states, “Solve. Show or explain your work. Jeff is making fruit punch. The recipe includes equal amounts of mango and pineapple juice and 490 mL of orange juice. If the recipe makes 1 L 50 mL, how much of each of the mango and pineapple juices are in the recipe?” (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. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.)

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

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 18, Anchor Task, Problem 3, students develop conceptual understanding and application as they add and subtract whole numbers. The problem states, “There were 12,345 people at a concert on Saturday night. On Sunday night, there were 1,795 fewer people at the concert than on Saturday night. How many people attended the concert on both nights?” Guiding Questions for teachers include, “How can we use a tape diagram to solve? What will it look like? How can we write an equation (or equations) to represent this problem, using a letter to represent the unknown? Did anyone assess reasonableness as they were solving the problem, before being asked to do so? Did you use estimation or the relationship between addition and subtraction to assess reasonableness?” (4.NBT.4: Fluently add and subtract multi-digit whole numbers using the standard algorithm.)

• In Unit 6, Decimal Fractions, Lesson 12, Target Task, students develop procedural skill and  fluency and application as they solve word problems using fractions. The task states, “Solve. Show or explain your work. Jay has \frac{29}{100}L of oil.  He buys another 1\frac{5}{10}L.  He still needs \frac{71}{100}L to deep fry clams. How much oil does Jay need to make fried clams?” (4.NF.5: Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.)

• In Unit 7, Unit Conversions, Lesson 12, Problem Set, Problem 7, students develop conceptual understanding and application as they solve word problems involving time. The problem states, “A cartoon lasts \frac{1}{2} hour. A movie is 6 times as long as the cartoon. How many minutes does it take to watch both the cartoon and the movie? If the cartoon started at 11:04 am and the movie came on immediately after the cartoon, what time will the movie be over?” Discussion of Problem Set questions include, “How many different ways were 7 halves represented? (30 min 7, as \frac{7}{2} and as \frac{6}{2}+\frac{1}{2}) What advantage is there to knowing all of these representations when it comes to solving a problem like this one? What shortcuts or efficiencies did you use today when solving your problems? How do you decide whether to start by converting to a smaller unit or to work with the mixed number or decimal measurements?” (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.)

he materials reviewed for Fishtank Plus Math Grade 4 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. MPs are explicitly identified for teachers within the Unit Summary and specific lessons (Criteria for Success, Tips for Teachers, or Anchor Task notes).

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:

• In Unit 1, Place Value, Rounding, Addition and Subtraction, Lesson 18, Homework, Problem 1, students “Assess the reasonableness of answers by choosing a place to round the given values to, rounding them, and using those to compute the solution or using the relationship between addition and subtraction to check answers.” The problem states, “Zachary’s final project for a college course took a semester to write and had 95,234 words. Zachary wrote 35,295 words the first month and 19,240 words the second month. a. Round each value to the nearest ten thousand to estimate how many words Zachary wrote during the remaining part of the semester. b. Find the exact number of words written during the remaining part of the semester. c. Use your answer from (a) to explain why your answer in (b) is reasonable.”

• In Unit 2: Multi-Digit Multiplication, Lesson 18, Target Task, students “Assess the reasonableness of an answer (MP.1).” The problem states, “Solve. Show or explain your work. Michael earns $9 per hour. He works 28 hours each week. David earns $8 per hour. He works 40 hours each week. After 6 weeks, who earns more money? How much more money?”

• In Unit 3, Multi-Digit Division, Lesson 12, Anchor Task, Problems 1 and 2, with teacher support, students “Make sense of a three-act task and persevere in solving it (MP.1);” “Interpret the remainder in the context of a problem (MP.1);” and “Assess the reasonableness of an answer (MP.1).” The problem states, “Act 1: Look at the following image [image of a boy sitting reading a book]. How long will it take him to finish the book?” Guiding questions include, “What do you notice? What do you wonder? Which of our noticings and wonderings is the question we most want the answer to? (The question you want to zero in on is stated in the task above.) Make an estimate of the answer. What information do you need to figure out the answer?”  Problem 2 continues with Act 2: “Use the following information to solve - Diary of a Wimpy Kid is 221 pages long. Joao has already read 15 pages. He only reads during independent reading and reads nine pages during each block.” Guiding questions include, “Do you have all the information you need to solve?” 

• In Unit 5, Fraction Operations, Lesson 3, Anchor Tasks, Problem 4, students “Assess the reasonableness of answers based on estimates (MP.1).” The problem states, “First, estimate the following solutions. Then add or subtract. a. \frac{3}{4}+\frac{1}{4} b. 1-\frac{3}{10}.” Guiding questions include, “What do you expect the solution to be? Why? How can we think of this in unit form? How will that help us solve? What picture can we draw to model the problem? Can we rewrite our solution using larger units? Based on our estimate, is the computed solution reasonable? Why or why not? What did we learn about adding and subtracting like units in Anchor Task #1? What does that tell us about how we can rewrite Part (b) to solve?”

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:

• In Unit 2, Multi-Digit Multiplication, Lesson 2, Anchor Tasks, Problem 3, students “Write an equation to represent a multiplicative comparison word problem with a smaller unknown, using a letter to represent the unknown (MP.2).” The problem states, “Ella weighs 36 pounds. Her brother, Farid, weighs four times less than her. How many pounds does Farid weigh? Represent the situation as an equation to solve.” Guiding questions include, “What is different about the way this question is phrased from the ones we’ve seen so far in this lesson and the previous one? How can we represent this situation with a tape diagram? Where do you see ‘four time less than’ in the tape diagram? How can we represent this situation with an equation? How can you use a letter to represent the unknown? Where do you see ‘four times less than’ in the equation?” 

• In Unit 6, Decimal Fractions, Lesson 3, Anchor Task, Problem 1, students represent decimals to tenths using pictorial base ten blocks and convert between fraction, decimal, unit, and fraction and decimal expanded form. “Below are representations for a ten, a one, and a tenth. Based on the base ten block diagrams below, fill in the table with the value of each diagram (a)-(c).” Notes, “This task is an opportunity for students to reason abstractly and quantitatively (MP.2). The base ten blocks allow students to see how many copies of each unit there are in a value and use the idea of copies to translate each value to a multiplication expression, making the connection between the number, its representation, and its expanded forms more apparent.”

• In Unit 8, Shapes and Angles, Lesson 13, Anchor Tasks, Problem 1, students, “Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems that involve more than two adjacent angles, e.g., by using an equation with a symbol for the unknown angle measure (MP.2, MP.4)” and “Write an equation to represent the unknown angle measure in a compound angle (MP.2).” The problem states, “Use patterns blocks of various types to create a design in which you can see a decomposition of 360°. Which shapes did you use? Write an equation to show how you composed 360°.” Guiding Questions include, “What is the angle measure of each angle that you used to compose the 360° angle? Write a number sentence to represent the relationship between the angle measure of each of the pattern blocks and the total angle measure. (Remove a shape from the composed figure.) Let’s say we didn’t know the size of the missing angle. How could we find it? How can we use a letter to represent our unknown in the number sentence?”

The materials reviewed for Fishtank Plus 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. MP3 is explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes) and students engage with the full intent of the MP through a variety of lesson problems and assessment items.

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 Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 12, Anchor Tasks, Problem 1, students “Determine the most appropriate place value to round to to have a reasonable estimate, based on the context of the problem.” The problem states, “In the year 2015, there were 935,292 visitors to the White House. They each got a map to guide them around. President Obama said that they can round to the nearest thousand to decide about how many maps to order for next year. Do you agree or disagree with President Obama? Explain.” Guiding Questions include, “What does it mean for an answer or estimate to be reasonable? Was President Obama’s estimate reasonable? Why or why not? Let’s round 935,292 to every place. What happens as we round it to smaller and smaller places? Of all of the possible rounded values, which one would have been best for President Obama to use? Why?” Note for the teacher include, “There isn’t necessarily a right answer to the last question, especially since a past figure is being used to predict a future one. But answers that are probably not correct are those that result in an estimate that is lower than the actual figure (like Obama’s).”

• In Unit 3, Multi-Digit Division, Lesson 4, Anchor Tasks, Problem 1, students “understand and explain why various mental strategies work.” The problem states, “Three different students solved the problem 232 ÷ 8 below. Study their work and determine whether the strategy they used works.” Three student examples are provided. Guiding questions include, “How does each visual representation show a solution to 232 ÷ 8? Can you write an equation to represent the way each student thought about computing 232 ÷ 8? Look at Student A and Student B. How are their strategies similar? How are they different? What is another way you could have broken up 232 into chunks to divide? How is what Student C similar to Student A and Student B? How is it different? Are there other ways of solving 232 ÷ 8  that you can think of that aren’t represented here?”

• In Unit 6, Decimal Fractions, Lesson 7, Target Task, Problem 2, students “justify a comparison using a visual model or reasoning.” The problem states, “Ryan says that 0.6 is less than 0.60 because it has fewer digits. Jessie says that 0.6 is greater than 0.60 . Who is right? Why? Show or explain your work.”

The materials reviewed for Fishtank Plus Math Grade 4 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. The MPs are explicitly identified for teachers within the unit summary or specific lessons (Criteria for Success, Tips for Teachers, or Anchor Tasks).

MP4: Model with mathematics, is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students are given many opportunities to solve real-world problems, identify important quantities to make sense of relationships, and represent them mathematically. They model with math as they work with support of the teacher and independently throughout the units. Examples include:

• In Unit 2, Multi-Digit Multiplication, Lesson 17, Target Task, students “Solve two-step word problems involving multiplication, addition, and subtraction (MP.4).” The task states, “Solve. Show or explain your work. Jennifer has 256 beads. Stella has 3 times as many beads as Jennifer. Tiah has 104 more beads than Stella. How many beads does Tiah have?”

• In Unit 3, Multi-Digit Division, Lesson 1, Anchor Tasks, Problem 1, students “Solve division word problems within 100 that involve a remainder, using an array, an area model, or a tape diagram to represent the problem (MP.4).” The problem states, “There are 13 students to be split among 4 teams. How many students will be on each team?” Guiding Questions include, “How could you represent this problem with an array? An area model? A tape diagram? A remainder is the number left over when one number is divided by another. What is the quotient in this problem? What is the remainder?”  

• In Unit 5, Fraction Operations, Lesson 19, Anchor Task, Problem 1, students solve problems involving multiplication of fractions. Tips for Teachers include, “Today’s three-act task helps illustrate Mathematical Practice Standard 4, Model with mathematics. Students apply the mathematics they know to solve problems arising in everyday life. As students think about the situation in real life, they have to select the mathematical thinking that will help them solve the problem. This selection of a mathematical model allows students to think about which mathematics might be useful in the real-life situation that they face. Students identify the mathematical elements of the situation and decide which solution pathway is best for them to follow.”  Problem 1 states, “Act 1: Watch the following video: ‘How Much Sugar.’ How much sugar is in the entire pack of soda cans?” Guiding Questions include, “What do you notice? What do you wonder? Which of our noticings and wonderings is the question we most want the answer to? (The question you want to zero in on is stated in the task above.)  Make an estimate of the answer. What information do you need to figure out the answer?” 

MP5: Use appropriate tools strategically, 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 throughout the materials, with support of the teacher or during independent practice, to identify and use a variety of tools or strategies that support their understanding of grade level math. Examples include:

• In Unit 3, Multi-Digit Division, Lesson 3, Anchor Tasks, Problem 2, students engage with MP5 “as they illustrate and explain the calculation by using physical or drawn models, they are using appropriate drawn tools strategically (MP.5) and attending to precision (MP.6) as they use base-ten units in the appropriate places (PARCC Model Content Frameworks, Math, Grades 3–11).” The problem states, “Solve. a. 32 ÷  4, b. 320 ÷  4, c. 3,200 ÷  4.” 

• In Unit 4, Fraction Equivalence and Ordering, Lesson 11, Anchor Tasks, Problem 1, students “Compare two fractions with the same numerators or denominators using visual models.” The problem states, “Below are measurements of ribbon in feet. For each pair of ribbons, determine which one is longer. Show or explain how you know. a. \frac{3}{4} ft. and \frac{1}{4} ft. b. \frac{5}{12} ft. and \frac{5}{6} ft.” Guiding Questions include, “What’s more, 1 apple or 3 apples? How is this related to the idea of comparing 1 fourth and 3 fourths? How can you represent both measurements in Part (a) with a tape diagram? With a number line? How can you use those representations to determine which measurement is larger? Record the comparison with the correct symbol. (Write \frac{3}{4}>\frac{1}{4}.) What’s more, 1 dollar or 1 cent? What about 5 dollars or 5 cents? What’s more, 1 sixth or 1 twelfth? How is this related to the idea of comparing 5 twelfths with 5 sixths? How can you represent both measurements in Part (b) with a tape diagram? With a number line? How can you use those representations to determine which measurement is larger? Write a number sentence to record this comparison. (Write \frac{5}{12}<\frac{5}{6}.)  How would our answer to (a) change if the first ribbon were \frac{3}{4} inch? What about our answer to (b) if the second ribbon were \frac{5}{6} inch?”

• In Unit 8, Shapes and Angles, Lesson 1, Problem Set, Problem 3, students draw line segments with self-selected tools, “The students in Ms. Sun’s class were drawing geometric figures. First she asked them to draw some points, and then she asked them to draw all the line segments they could that join two of their points. a. Joni drew 4 points and then drew 4 line segments between them: [drawing provided] Are there other line segments that Joni could have drawn?” 

The materials reviewed for Fishtank Plus Math Grade 4 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. MP6 is explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes), and students engage with the full intent of the MP through a variety of lesson problems and assessment items.

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

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 11, Criteria For Success states, “Understand the advantages and disadvantages of rounding a value to various place values, including the precision of rounding a number to a smaller place value (MP.6) and ease of working/operating when rounding a number to a larger place value.” In Problem Set, Problem 8 states, “Look at Part (c) of #3-7. a. What do you notice about what 289,091 rounded to, depending on the place value? b. Which estimate is most precise? How do you know?”

• In Unit 2, Multi-Digit Multiplication, Lesson 16, Criteria for Success states, “Understand that when finding areas of rectangular regions answer will be in square units, and when finding the perimeter of a rectangular region, answers will be in linear units (MP.6).” In Homework, Problem 3, students understand that when finding areas of rectangular regions the answer will be in square units, and when finding the perimeter of a rectangular region, answers will be in linear units. The problem states, “The figure below represents a play space that Logan fenced in for his dog. Logan is getting a second dog and wants to increase the length of the play space by 3 feet and the width by 3 feet. What will be the difference in the area, in square feet, between the original play space and the new play space?”

• In Unit 6, Decimal Fractions, Lesson 7, Criteria For Success states, “Understand that decimal comparisons are only valid when the two decimals refer to the same whole (MP.6).” In Problem Set, Problem 4 states, “Danielle says that 0.17 is greater than 0.2 because 17 is greater than 2. Identify the incorrect reasoning in Danielle’s statement. Explain how Danielle can correct her reasoning. Use >, <, or = to give a correct comparison between 0.17 and 0.2.”

• In Unit 8, Shapes and Angles, Lesson 2, Criteria for Success states, “Draw right, obtuse, and acute angles (MP.5, MP.6).” In Target Task, Problem 2, students attend to precision as they independently create a variety of angles, “Draw an additional example of each type of angle mentioned above. a. Right angle; b. Obtuse angle; c. Acute angle; d. Straight angle.”

Students have frequent opportunities to attend to the specialized language of math in connection to grade-level content as they work with support of the teacher and independently throughout the units. The “Tips for Teachers” sections provide teachers with an understanding of grade-specific language and how to stress the specialized language during the lesson. Examples include:

• Each Unit Overview provides a link to a Fourth Grade Vocabulary Glossary. The glossary contains a chart with the columns “Word” and “Definition.” Under the Definition column, is the mathematical definition and an example. For example, for hundred thousand the definition reads, “Place value to the left of the ten thousands whose value is ten times as much as the ten thousands” and “Example - 100,000.”

• In Unit 4, Fraction Equivalence and Ordering, Lesson 10, Tips for Teachers states, “The term “simplify”/ “simplification” is intentionally excluded from CCSS since “it is possible to over-emphasize the importance of simplifying fractions in this way. There is no mathematical reason why fractions must be written in simplified form, although it may be convenient to do so in some cases” (NF Progression, p. 6). It is important for students to understand the term at some point, but because the focus of this unit is purely on equivalent fractions rather than some of those fractions being ‘simpler’ than others, the term is excluded from the unit. Instead, the conversation centers on the number and sizes of the parts, in keeping with the language of the standard (4.NF.1). Students are prompted to find fractions in the largest possible terms so that they have practice doing so for when it makes sense to do so, but they aren’t expected to do so in every task in the Anchor Tasks and Problem Set.”

• In Unit 6, Decimal Fractions, Lesson 1, Tips For Teachers state, “While ‘mathematicians and scientists often read 0.5 aloud as “zero point five” or “point five,”’ refrain from using this language until you are sure students have a strong sense of place value with decimals (NF Progression, p. 15).”

The materials reviewed for Fishtank Plus Math Grade 4 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. MPs are explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes).

MP7: Look for and make use of structure, 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 throughout the units to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently. Examples include:

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 1, Criteria for Success states, “Notice patterns in the count sequence, such as the repeating sequence of ones digits, that the tens digit changes every ten numbers, and the hundreds digit changes every 100 numbers, etc. (MP.7).” Homework, Problem 8 states, “Find 503 in your 1,000 book. a. Which two tens is it between? b. What two hundreds is it between? c. What would you have to change to make its neighbor above it?”

• In Unit 3: Multi-Digit Division, Lesson 14, Criteria For Success states, “Identify the rule of a growing number pattern (MP.7, MP.8).” Target Task states, “The first number in a pattern is 3. The pattern rule is to add 4. a. What is the seventh number in the pattern? 3, ___, ___, ___, ___, ___, ___ b. Explain why all of the terms in the pattern are odd.”

• In Unit 4, Fraction Equivalence and Ordering, Lesson 2, Criteria for Success states, “1. Look for structure (MP.7) to find patterns in multiples of various factors, such as a. All multiples of 2 are even numbers (i.e., end in 0, 2, 4, 6, or 8); b. All multiples of 3 have digits that add up to 3, 6, or 9; c. All multiples of 5 end in 0 or 5; d. All multiples of 6 end in 0, 2, 4, 6, 8 (i.e., are even/divisible by 2) and have digits that add up to 3, 6, or 9 (i.e., are divisible by 3); e. All multiples of 9 have digits that add up to 9 (including adding the digits of subsequent sums together, e.g., 99 → 9 + 9 = 18 and 1 + 8 = 9); f. All multiples of 10 end in 0. 2. Make use of structure (MP.7) by using the divisibility rules stated above to determine whether a number larger than 100 is a multiple of 2, 3, 5, 6, 9, or 10.” In Problem Set, Problem 2 states, “a. List the first 10 multiples of three below. b. What do you notice about the sum of the digits in each multiple? c. Use your observation in Part (b) to determine whether 582 is a multiple of 3.” 

• In Unit 6, Decimal Fractions, Lesson 5, Criteria for Success states, “Understand the value of each digit in a multi-digit decimal to hundredths, using pictorial base ten blocks to help (MP.7).” Homework, Problem 1, students understand the value of each digit in a multi-digit decimal to hundredths, using pictorial base ten blocks to help. The problem states, “Represent the values on the number line and area model. Convert the fraction to a decimal or vice versa. a. 2\frac{35}{100}= ____.____.” An area model and number line are provided. 

MP8: Look for and express regularity in repeated reasoning, 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 throughout the materials, with support of the teacher or during independent practice, to use repeated reasoning in order to make generalizations and build a deeper understanding of grade level math concepts. Examples include:

• In Unit 2, Multi-Digit Multiplication, Lesson 4, Criteria for Success states, “Identify patterns in multiplication of 10, 100, and 1,000 by one- and two-digit numbers (MP.8).” Anchor Tasks, Problem 1, students identify patterns in multiplication of 10, 100, and 1,000 by one- and two-digit numbers. The problem states, “1. Solve. a. 3 × 1 = ____; b. 3 × 10 = ____; c. 3 × 100 = ____;  d. 3 × 1,000 = ____;  2. What do you notice about #1? What do you wonder?” Guiding Questions include, “How can you use repeated addition or skip-counting to solve? How can you use unit form to solve? For example, what is 3 times 1 one? 3 times 1 ten? 1 hundred? 1 thousand? How is 3 x 10 related to 3 x 100? How can you use 3 x 10 to solve 3 x 100? (It may help to write this as 3 × 100 = 3 × (10 × 10) = (3 × 10) × 10 = 30 × 10 = 300, using parentheses to facilitate the use of the associative property.) How can you use ‘times as many’ to describe these equations?”

• In Unit 3, Multi-Digit Division, Lesson 3, Criteria for Success states, “Identify patterns in division of multiples of 10, 100, and 1,000 by single digits (MP.8).” In Homework, Problem 1, students identify patterns in division of multiples of 10, 100, and 1,000 by single digits. The problem states, “Find the quotient. Show or explain your work. a. 12 ÷ 3 = ___; b. 120 ÷ 3 =___; c. 1,200 ÷ 3 =___; d. How did the basic fact 12 ÷ 3 help you to solve Parts (b) and (c)?”

• In Unit 5, Fraction Operation, Lesson 17, Criteria for Success states, “Generate a general method for multiplying a whole number by a non-unit fraction, i.e., n × $$\frac{a}{b}$$ = \frac{n×a}{b} (MP.8).” In Homework, Problem 4, students generate a general method for multiplying a whole number by a non-unit fraction. The problem states, “Paloma is working on multiplying fractions and whole numbers 3 × \frac{4}{5} = 3 × 4 × \frac{1}{5} = \frac{12}{5} Is her work reasonable? Why or why not?”

• In Unit 7, Unit Conversions, Lesson 6, Criteria for Success states, “Use these relationships to convert measurements from a larger customary weight unit to a smaller unit (MP.7, MP.8). Use these relationships to convert measurements from mixed customary weight units to a smaller unit (MP.7, MP.8).” In Homework, Problem 4, students use the relationships between the backpacks to convert measurements from a larger customary weight unit to a smaller unit. The problem states, “The total weight of Sarah’s and Amanda’s full backpacks is 27 pounds. Sarah’s backpack weighs 15 pounds and 9 ounces. How much does Amanda’s backpack weigh?”