4th Grade - Gateway 2

The instructional materials for Match Fishtank Mathematics Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade-level, for example: 

• In Unit 4, Lesson 5, students create a wedge from folding a paper circle, then measure angles (4.MD.5) in wedges. In the Anchor Tasks, Problem 3, Guiding Questions state, “How can we use our wedge to measure the angles in Anchor Task 1? Why is it important to make sure there are no gaps or overlaps when we mark and move forward our wedges? How is this similar to other units of measurement like square units for area or length units for length?” 
• In Unit 5, Lesson 3, students draw area models to recognize and generate equivalent fractions (4.NF.1). In the Anchor Tasks, Problem 2 states, “Find two fractions that are equivalent to each of the following: a. $$\frac{1}{8}$$  b. $$\frac{5}{12}$$  c. $$\frac{4}{3}$$.” Guiding Questions state, “What can we draw to represent 1/8? How can we use that picture to find another equivalent fraction?”
• In Unit 8, Lesson 1, students explore the relationships between millimeters, centimeters, and meters on a meter stick (4.MD.1). In the Anchor Tasks, Problem 1 states, “Find the millimeter, centimeter, and meter measurement on the meter stick. a. What do you notice about the relationship between the length of a millimeter and the length of a centimeter? b. What do you notice about the relationship between the length of a centimeter and the length of a meter? c. What do you notice about the relationship between the length of a millimeter and the length of a meter?” Guiding Questions state, “How many times as large as a millimeter is a centimeter? How do you know? (Ask similar questions that correspond to parts (b) and (c).) We can’t use the meter stick to see the relationship between a meter and a kilometer. But, the prefix ‘kilo-’ means ‘thousand.’ Based on the meaning of the prefix, what do you think is the relationship between a kilometer and a meter? What are some things that are approximately a millimeter long? A centimeter? A meter? A kilometer? What is the length of your desk in millimeters? In centimeters? In meters? What do you notice about those values?”

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade, for example:

• In Unit 1, Lesson 10, students use a number line to understand the concept of rounding (4.NBT.3). In Anchor Task, Problem 1, students are given a blank number line and solve, “Is 4,125 closer to 4,000 or 5,000? Plot 4,000 and 5,000 on the two outermost spots on the number line below. Then plot 4,125 to prove your answer.” 
• In Unit 5, Lesson 2, students use number lines to generate equivalent fractions (4.NF.1). The Target Task states, “The fraction $$\frac{1}{4}$$ is represented on the number line below. Use the number line to find two more equivalent fractions.”
• In Unit 5, Lesson 8, students compare fractions with different denominators by using visual models to represent both fractions (4.NF.2). In the Anchor Tasks, Problem 1, students decide, “Would you rather have the leftover brownies in Scenario A or Scenario B? The pans in which the brownies were cooked are the same size.” Guiding Questions include but are not limited to, “What fraction of each brownie pan is left in Scenario A and Scenario B? How could we represent the common numerator or common denominator in each area model? Why does it matter that the brownie pans were the same size? How might your answer change if they weren’t the same size?”
• In Unit 6, Lesson 5, students “decompose non-unit fractions less than or equal to 2 as a sum of unit fractions and as a whole number times a unit fraction” (4.NF.3b,4a) through tape diagrams, number lines and various picture models in the Problem Set. In the Anchor Tasks, Problem 2, students “a. Represent the following fractions as a sum of unit fractions, a sum of non-unit fractions, and a multiple of a unit fraction. Write your answer as an equation and justify your equation with a tape diagram or number line. $$\frac{5}{3}$$  b. Use your work in part (a) to show why $$\frac{5}{3}=1\frac{2}{3}$$.”

The instructional materials for Match Fishtank Mathematics Grade 4 for attending to those standards that set an expectation of procedural skill and fluency.

The structure of the lessons includes several opportunities to develop these skills, for example:

• In the Unit Summary, procedural skills for the unit are identified.
• Throughout the materials, Anchor Tasks provide students with a variety of problem types to practice procedural skills.
• Problem Sets provide students with a variety of resources or problem types to practice procedural skills.
• There is a Guide to Procedural Skills and Fluency under teachers tools and mathematics guides. 

The instructional materials develop procedural skill and fluency throughout the grade level. The instructional materials provide opportunities for students to demonstrate procedural skill and fluency independently throughout the grade level, especially where called for by the standards (4.NBT.4). For example:

• In Unit 1, Lesson 13, students “fluently add multi-digit whole numbers using the standard algorithm (4.NBT.4) involving up to two compositions. Solve one-step word problems involving addition.’ In Anchor Tasks, Problem 2, students “Estimate. Then solve. a. $$40,762 + 30,473 =$$___  b. ___ $$= 258,209 + 48,906$$.”
• In Unit 1, Lesson 14, students fluently add multi-digit whole numbers using the standard algorithm (4.NBT.4). Students have opportunities to practice in the Problem Set, Homework, and Target Task. For example, Problem Set, Problem 1 states, “Solve. a. $$6,314 + 2,493$$   b. $$8,314 + 2,493$$ c. ____$$= 12,378 + 5,463$$ d. $$52,098 + 6,048 $$ e.____ $$- 34,698 = 71,840$$ f. $$544,811 + 356,445$$ g. $$527 + 275 + 752 = g$$ h. $$478,295 + 353,067$$ i.____ $$= 38,193 + 6,376 + 241,457$$.”
• In Unit 1, Lesson 16, students “fluently subtract multi-digit whole numbers using the standard algorithm involving two decompositions” (4.NBT.4). In the Target Task, students “Solve, Show or explain your work. a. $$8,512 - 2,501$$  b. $$18,042 - 4,122$$ c. $$19,850 - 15, 761$$.”
• In Unit 1, Lesson 19, students solve multi-step word problems involving addition and subtraction,” (4.OA.3, 4.NBT.4). In the Homework, Problem 3 states, “There were 22,869 children, 49,563 men, and 2,872 more women than men at the fair. How many people were at the fair?”
• In Unit 2, Lesson 23, students “solve multi-step word problems involving multiplication, addition, and subtraction,” (4.OA.3, 4.NBT.4). The Target Task 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, Lesson 3, students “divide multiples of 10, 100, and 1,000 by one-digit numbers,” (4.NBT.6). Anchor Tasks, Problem 1 states, “1. Solve. a. $$9\div3$$   b. $$90\div3$$  c. $$900\div3$$   d. $$9,000\div3$$.  2. What do you notice about #1? What do you wonder?”

The instructional materials for Match Fishtank Mathematics Grade 4 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

In Problem Sets and Target Tasks, students engage with real-world problems and have opportunities for application, especially where called for by the standards (4.OA.3, 4.NF.3d, 4.NF.4c). The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge. Students have opportunities to independently demonstrate the use of mathematics flexibly in a variety of contexts. Examples of routine application include, but are not limited to:

• In Unit 2, Lesson 23, students solve multi-step, real-world problems involving multiplication, addition and subtraction (4.OA.3). For example, Problem Set, Problem 4 states, “There will be 45 adults going to a museum. There will be twice as many students as adults. Adult tickets cost $25 each. Students tickets cost $12 each. What is the total cost for the students and adults?” 
• In Unit 3, Lesson 11, students apply the formulas for area and perimeter in real-world and mathematical problems involving all four operations (4.OA.3, 4.MD.3). For example, Problem Set, Problem 3 states, “Mr. Felton will use exactly 42 feet of fencing to surround a garden that is in the shape of a rectangle. His garden has a length of 12 feet. The equation below represents the perimeter of Mr. Felton’s garden. $$w + w + 12 + 12 = 42$$ What is the width, in feet, of Mr. Felton’s garden?”
• In Unit 3, Lesson 13, students solve multi-step word problems involving the four operations (4.OA.3). For example, Problem Set, Problem 2 states, “Your class is collecting bottled water for a service project. The goal is to collect 400 bottles of water. Sarah wheels in 6 packs, each containing 6 bottles of water. Sarah brought in twice as many packs as Max. After counting Sarah and Max’s bottles, how many packs of water still need to be collected?” 
• In Unit 6, Lesson 17, students solve word problems involving addition and subtraction of fractions (4.NF.3d). For example, Target Task, Problem 1 states, “Mrs. Cashmore bought a large melon. She cut a piece that weighed $$1\frac{1}{8}$$ pounds and gave it to her neighbor. The remaining piece of melon weighed $$2\frac{5}{8}$$ pounds. How much did the whole melon weigh?” 
• In Unit 6, Lesson 20, students solve word problems involving multiplication of fractions (4.NF.4c). For example, Target Task, Problem 2 states, “If a bucket holds $$2\frac{1}{2}$$ gallons and 43 buckets fill a tank, how much does the tank hold?”

Examples of non-routine application include, but are not limited to:

• In Unit 2, Lesson 23, students solve multi-step word problems involving multiplication, addition, and subtraction (4.OA.3). For example, Problem Set, Problem 8 states, “The table below shows the number of tickets sold at a movie theatre on Friday. (table provided) The number of each type of ticket sold on Saturday is described below. Adult tickets - 2 times as many as the number of adult tickets sold on Friday. Children’s tickets - 3 times as many as the number of children’s tickets sold on Friday. Complete the table above to show the numbers of tickets sold on Saturday. What is the total number of tickets sold over these two days?”
• In Unit 6, Lesson 21, students solve word problems involving addition, subtraction, and multiplication of fractions (4.NF.3d and 4.NF.4c). For example, Target Task states, “The table below shows the sizes and weights of containers of potato salad sold at a store. Kim purchased 6 small containers of potato salad and Seth purchased 2 extra-large containers of potato salad. What is the difference in the weights, in pounds, of Kim's and Seth's purchases?”
• In Unit 7, Lesson 12, students solve word problems involving the addition of decimals and decimal fractions (4.NF.5). For example, in Problem Set, Problem 6 states, “A team of three ran a relay race. The final runner’s time was the fastest, measuring 29.2 seconds. The middle runner’s time was 1.89 seconds slower than the final runner’s. The starting runner’s time was 0.9 seconds slower than the middle runner’s. What was the team’s total time for the race?”
• In Unit 8, Lesson 4, students solve multi-step, real-world problems that require metric unit conversions of length, mass, and capacity (4.MD.2). For example, in Problem Set, Problem 4 states, “Enya walked 2km 309m from school to the store. Then, she walked twice that amount from the store back home. How far, in meters, did she walk in total?”
• In Unit 8, Lesson 12, students solve word problems involving converting fractional and decimal measurements to a smaller unit. (4.MD.2). For example, in Problem Set, Problem 2 states, “Five ounces of pretzels are put into each bag. How many bags can be made from 22 3/4 pounds of pretzels?”

The instructional materials for Match Fishtank Mathematics Grade 4 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present in the instructional materials. Many of the lessons incorporate two aspects of rigor with an emphasis on application. Student practice includes all three aspects of rigor, though there are fewer questions for conceptual understanding. There are instances where all three aspects of rigor are present independently throughout the program materials. 

Examples of Conceptual Understanding include:

• In Unit 1, Lesson 9, students use number lines to locate multi-digit numbers and explain their placement (4.NBT.2). For example, Anchor Task, Problem 1 states, “Based on where 0 and 100 are, what number do you think the question mark is on? Explain your choice.” 
• In Unit 5, Lesson 2, students “recognize and generate equivalent fractions with smaller units using number lines (4.NF.1).” Anchor Task, Problem 1 states, “a. What fraction represents the point shown on the number line below? B. What fraction represents the point shown on the number line below? C. Use the pictures to explain why the two fractions represented above are equivalent.”
• In Unit 6, Lesson 5, students “decompose non-unit fractions less than or equal to 2 as a sum of unit fractions and as a whole number times a unit fraction (4.NF.3,4).” Anchor Task, Problem 1 states, “During Lesson 1, Francisco and Harry were playing with their fraction strips. Part A: They filled the outline as follows: What fraction of the original outline is each piece? Part B: Then Francisco and Harry build another shape, as shown below: What fraction of the original outline is this new shape? Explain your answer.”

Examples of Procedural Skills and Fluency include:

• In Unit 1, Lesson 13, students “Fluently add multi-digit whole numbers using the standard algorithm involving up to two compositions. Solve one-step word problems involving addition,” (4.NBT.4). In Problem Set, Problem 3 states, “Elizabeth solved the problem $$70,912 + 24,628$$ using the standard algorithm, as shown below. a. Identify the mistake in Elizabeth’s work. b. Show or explain how Elizabeth can correct her mistake.”
• In Unit 1, Lesson 17, students “Fluently subtract multi-digit whole numbers using the standard algorithm involving multiple decompositions. Solve one-step word problems involving subtraction,” (4.NBT.4). In Anchor Task, Problem 1 states, “Place any digit, 1 through 9, in the boxes below to create the smallest possible difference. Each digit can only be used once.”
• In Unit 2, Lesson 6, students find factor pairs for numbers to 100 (4.OA.3,4). For example, Problem Set, Problem 2 states, “Find all factors for the following numbers.” Students complete tables of the factor pairs for 25, 28, and 29. 

Examples of Application include:

• In Unit 2, Lesson 22, students “solve two-step word problems involving multiplication, addition, and subtraction and assess the reasonableness of answers,” (4.OA.2,3). For example, the Target 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 6, Lesson 20, students “solve word problems involving multiplication of fractions,” (4.NF.4c). In Anchor Task, Act 2 states, “Share the following information with students:  One can of soda has $$\frac{4}{15}$$ cup of sugar in it. There are 12 cans in the whole pack. Have students work on the task to determine how much sugar is in the whole pack.”
• In Unit 8, Lesson 4, students “solve multi-step word problems that requires metric unit conversions of length, mass, and capacity,” (4.MD.2). For example, Problem Set, Problem 2 states, “Kiera likes to mix apple juice with cranberry juice in her glass. She has 1 L of juice in all. She puts three times as much apple juice as cranberry juice in her glass. How many mL of apple juice did she put in her glass?”

Examples of multiple aspects of rigor engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study include:

• In Unit 1, Lesson 16, students, “Fluently subtract multi-digit whole numbers using the standard algorithm involving up to two decompositions. (4.NBT.4) Students also, “solve one-step word problems involving subtraction.” For example, Problem Set, Problem 2 states, “Chuck’s mom spent $19,155 on a new car. She had $30,067 in her bank account. How much money does Chuck’s mom have after buying the car? Show or explain your work.” 
• In Unit 2, Lesson 10, students use their conceptual understanding of rounding to estimate multi-digit products when multiplying multiples of 10, 100, and 1,000 by one-digit numbers (4.NBT.5). For example, Problem Set, Problem 4 states, “Estimate the following products.” In Problem 4k, students solve the following problem, “$$3\times9,268$$.”
• In Unit 6, Lesson 17, students use their conceptual understanding of addition and subtraction to “solve word problems involving addition and subtraction of fractions,” (4.NF.3d). For example, the Target Task states, “Mrs. Cashmore bought a large melon. She cut a piece that weighed $$1\frac{1}{8}$$ pounds and gave it to her neighbor. The remaining piece of melon weighed $$2\frac{5}{8}$$ pounds. How much did the whole melon weigh?”

The instructional materials reviewed for Match Fishtank Grade 4 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade-level.

All Standards for Mathematical Practice are clearly identified throughout the materials in numerous places, that include but are not limited to: Unit Summaries, Criteria for Success, and Tips for Teachers. Examples include:

• In Unit 3, Unit Summary, “Throughout the unit, students are engaging with the mathematical practices in various ways. For example, students are seeing and making use of structure (MP.7) as they ‘decompos[e] the dividend into like base-ten units and find the quotient unit by unit”’(NBT Progressions, p. 16). Further, "by reasoning repeatedly (MP.8) about the connection between math drawings and written numerical work, students can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities” (NBT Progression, p. 14). Lastly, as students solve multi-step word problems involving addition, subtraction, and multiplication, they are modeling with mathematics (MP.4).”
• In Unit 5, Lesson 4, Criteria for Success states, “2. Understand how the numbers and sizes of parts differ even though the two fractions are the same size, and connect this idea to the general method of using multiplication to find an equivalent fraction (MP.7). 4. Determine whether two fractions are equivalent using multiplication, and support that work with a visual model (MP.3, MP.5).”
• In Unit 8, Lesson 5, Tips for Teachers, “As mentioned in the Progressions, “relating units within the traditional system provides an opportunity to engage in mathematical practices, especially ‘look for and make use of structure’ (MP.7) and ‘look for and express regularity in repeated reasoning’ (MP.8).”

Examples of the MPs being used to enrich the mathematical content include:

• MP7 is connected to the mathematical content in Unit 2, Lesson 5, Target Task, Problem 1, as students “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.” For example, “What do all multiples of 9 have in common?”
• MP4 is connected to the mathematical content in Unit 3, Lesson 8, Problem Set, Problem 4, as students “Solve one-step division word problems, including those that require the interpretations of the remainder.” For example, “Zach filled 581 one-liter bottles with apple cider. He distributed the bottles to 4 stores. Each store received the same number of bottles. Zach kept the leftover bottles to be distributed the next day. How many bottles did Zach keep?”
• MP6 is connected to the mathematical content in Unit 4, Lesson 2, Anchor Task, Problem 3, as students “Draw right, obtuse, and acute angles.” For example, “Draw an example of a right angle, an acute angle, an obtuse angle, and a straight angle.”
• MP4 is connected to the mathematical content in Unit 8, Lesson 8, Target Task, Problem 2, as students “solve one-step word problems that require time unit conversions, including problems that involve elapsed time (MP.4)” For example, “Jacob needs to do his chores. It takes Jacob an hour and 45 minutes to mow the lawn and 20 minutes to clean his room. If he starts his chores at 2:00 PM, what time will he finish?”

The instructional materials reviewed for Match Fishtank Mathematics Grade 4 meet expectations for carefully attending to the full meaning of each practice standard. 

The materials attend to the full meaning of each of the 8 Mathematical Practices (MPs). The MPs are discussed in both the nit and Lesson Summaries as they relate to the overall content. The MPs are also explained, when applicable, within specific parts of each lesson, including but not limited to the Criteria for Success and Tips for Teachers. Each practice is addressed multiple times throughout the year. Over the course of the year, students have ample opportunity to engage with the full meaning of every MP. Examples include but are not limited to:

• MP1: In Unit 3, Lesson 2, Criteria for Success states, “Interpret the remainder in the context of the word problem, including: c. Having the remainder be the answer to the problem (such as when wanting to know how much Halloween candy the teacher gets to keep after distributing some to students) (MP.1).” For example, Anchor Task, Problem 1 states, “Solve the following problems. Think about the main differences between each one. a. A teacher has 21 batteries. Each calculator uses 4 batteries. How many calculators can the teacher fill with batteries? b. Four children can ride in a car. How many cars are needed to take 21 children to the museum? c. Ms. Cole wants to share 21 pieces of candy with 4 students. If Ms. Cole gets to eat the pieces of candy that can’t be split evenly, how many pieces of candy will Ms. Cole get?” 
• MP2: In Unit 2, Lesson 21, Tips for Teachers states, “When engaging in the mathematical practice of reasoning abstractly and quantitatively (MP.2) in work with area and perimeter, students think of the situation and perhaps make a drawing. Then they recreate the ‘formula’ with specific numbers and one unknown number as a situation equation for this particular numerical equation. (GM Progression, p. 22)” For example, Problem Set, Problem 4 states, “The rectangular projection screen in the school auditorium is 5 times as long as 5 times as wide as the rectangular screen in the library. The screen in the library is 9 feet long and 12 feet wide. What is the perimeter of the screen in the auditorium?”
• MP4: In Unit 8, Lesson 1, Criteria for Success states, “Solve one-step word problems that require metric length unit conversions (MP.4).”  For example, Anchor Task, Problem 3 states, “Mr. Duffy wants to buy a new pair of skis. Skis are sold based on their length in centimeters. If Mr. Duffy wants to buy a pair of skis that are his exact height, and he is 1 meter 83 centimeters tall, what length skis should Mr. Duffy buy? How can we determine what length skis Mr. Duffy should buy? Write an equation to represent the computation. (Write “1 m 83 cm = 1 x (1 m) + 83 x (1 cm) = 1 x (100 cm) + 83 x (1 cm) = 100 cm + 83 cm = 183 cm.”) How many centimeters is 5 m 4 cm? How many millimeters is 3 m 75 mm? How many centimeters is 20 km 45 m?”
• MP5: In Unit 4, Lesson 14, Criteria for Success states, “1. Classify triangles according to their side length, and understand each type as equilateral (has all three sides of equal length), isosceles (has two sides of equal length), and scalene (has no sides of equal length) (MP.2, MP.5). 2. Classify triangles according to their angle measure and understand each type as acute (has all three angles that are acute), right (has a right angle), and obtuse (has an obtuse angle) (MP.2, MP.5).” For example, in Problem Set, Problem 1a, students are prompted to choose their own tool to determine right triangles, “Which of these polygons are right triangles. Choose a measuring tool to help you determine this.”
• MP6: In Unit 1, 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 the Target Task, Problem 2 states, “There are 16,850 Star coffee shops around the world. Round the number of shops to the nearest thousand and ten thousand. Which answer is more precise? Explain your thinking using pictures, numbers or words.”
• MP7: In Unit 3, Lesson 3, Tips for Teachers states, “Throughout Lesson 3-10, students are seeing and making use of structure (MP.7) as they ‘decompose the dividend into like base-ten units and find the quotient unit by unit’ (Progressions for the Common Core State Standards in Mathematics, Number and Operations in Base Ten, K-5, p. 16). (4.NBT.6)” For example, Problem Set, Problem 4 states, “What is the missing number in the equation below? $$5,600\div8=?$$ a. 7 b. 70 c. 700 d. 7,000”
• MP8: In Unit 3, Lesson 3, Criteria for Success states, “Identify patterns in division of multiples of 10, 100, and 1,000 by single digits (MP.8).” For example, Anchor Task, Problem 2, students solve, “a. $$32\div4$$ b. $$320\div4$$ c. $$3200\div4$$.” Students then answer, “What do you notice about #1? What do you wonder?” 

The instructional materials reviewed for Match Fishtank Mathematics Grade 4 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

The student materials consistently prompt students to construct viable arguments and analyze the arguments of others, for example:

• In Unit 3, Lesson 4, Problem Set, Problem 2 states, “Jillian says, I know that 20 times 7 is 140 and if I take away 2 sevens that leaves 126. So $$126\div7 = 18$$. a. Is Jillian’s calculation correct? Explain. b. Draw a picture showing Jillian’s reasoning. c. Use Jillian’s method to find $$222\div6$$.” Students “Understand and explain why various simplifying strategies work (MP.3),” when dividing two-, three-, and four-digit numbers by one-digit numbers. 
• In Unit 4, Lesson 4, Problem Set, Problem 3 states, “Ms. Glynn said that the line segments that make up the E are parallel. Is she correct? Why or why not?” Students construct a viable argument in order to respond to the word problem (MP.3).
• In Unit 5, Lesson 4, Target Task, Problem 1 states, “Determine if the following fractions $$\frac{5}{3}=\frac{15}{9}$$ are equivalent. Then explain how you know.” Students construct an argument by determining if the given fractions are equivalent, then justify their conclusions (MP.3). 
• In Unit 6, Lesson 10, Problem Set, Problem 5 states, “Simone changed the mixed number 4 1/3 to a fraction. First, Simone change the whole number 4 to the fraction 4/3. Then she added the two fractions together. Her work is shown. $$4\frac{1}{3}=4+\frac{1}{3}=\frac{4}{3}+\frac{1}{3}=\frac{5}{3}$$  Explain the error in Simone’s reasoning. Find the correct equivalent fraction. Describe another method you can use to change the mixed number $$4\frac{1}{3}$$ to a fraction.” Students analyze the reasoning of others (MP.3). 
• In Unit 7, Lesson 7, Anchor Task, Problem 2 states, “Fill in the blank with <, =, or > to complete the equation. Justify your comparison.  0.8 and 0.3, 0.01 and 0.11, 0.2 and 0.20, 0.6 and 0.41, 0.07 and 0.70, 0.57 and 0.75.” Students justify their reasoning of comparisons of decimal equations (MP.3)

The instructional materials reviewed for Match Fishtank Mathematics Grade 4 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The teacher materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others through the Criteria for Success, Guiding Questions, and Tips for Teachers, for example:

• In Unit 2, Lesson 11, Criteria for Success states, “2. Understand and explain why various simplifying strategies work (MP.3).” In Problem Set, Problem 2 states, “Connor solves $$8\times16$$. He says, ‘I can find the product if I multiply $$8\times15$$ and then add 8.’ Select the statement that best explains if Connor’s strategy is correct. a. Connor is correct, because he can change the 16 to use an easier number to multiply by, like 15. b. Connor is incorrect, because $$8\times16$$ is the same as 4 groups of 8, but 4 groups of 8. c. Connor is correct, because $$8\times16$$ is the same as 15 groups of 8, plus 1 group of 8. d. Connor is incorrect, because he should add 16 instead of 8.” The Discussion of the Problem Set provides teachers with the following questions to support students constructing viable arguments: “What was the best option for #2? Were there other options that were decent options but not the best option? How did you decide which was the better option of the two?”
• In Unit 3, Lesson 10, Criteria for Success states, “3. Critique the reasoning of others regarding any of the above cases (MP.3).” Anchor Task, Problem 1, Guiding Questions include, but are not limited to, “How do you know that Geraldo’s answer is wrong? What was the error that Geraldo made? What is the correct answer? How did you solve? Is our answer reasonable? How can we check our work?”
• In Unit 5, Lesson 8, Criteria for Success states, “1. Compare two fractions where both fractions need to be replaced with an equivalent one to be able to compare common units (denominators) or common number of units (numerator) (MP.3).” In Anchor Task, Problem 1 states, “Would you rather have the leftover brownies in Scenario A or Scenario B? The pans in which the brownies were cooked are the same size.” Guiding Questions include, but are not limited to, “Why does it matter that the brownie pans were the same size? How might your answer change if they weren’t the same size?”
• In Unit 6, Lesson 13, Tips for Teachers states, “Throughout Lessons 13–17, “calculations with mixed numbers provide opportunities for students to compare approaches and justify steps in their computations (MP.3)” (NF Progression, p. 12).”
• In Unit 7, Lesson 6, Anchor Task, Problem 1 notes, “This task provides an opportunity to construct viable arguments and critique the reasoning of others (MP.3).”  Guiding Questions include, but are not limited to, “What place value unit does a one-dollar bill represent? A dime? A penny? What would 4 one-dollar bills and 1 dime be in pennies? How can we express that value using place value units? What about 42 dimes? If you wanted to represent 42 dimes using the largest possible monetary units, how could we do that? What about 413 pennies?”

The instructional materials reviewed for Match Fishtank Mathematics Grade 4 meet expectations for explicitly attending to the specialized language of mathematics.

Examples of the materials providing explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols include: 

• In Unit 1, Lesson 7, Tips for Teachers states, “Remember from Lesson 4 that when saying or writing a number in word form, the word ‘and’ implies a decimal point and therefore should not be used when naming whole numbers. For example, 217,350 is read ‘two hundred seventeen thousand three hundred fifty,’ not ‘two hundred and seventeen thousand three hundred and fifty.’ Even though students have not yet seen decimals, it is important to read numbers correctly before they do.”
• In Unit 2, Lesson 5, Tips For Teachers states, “Note that the lesson, including the objective, does not contain the language of ‘divisible’ or ‘divisibility.’ This is because it is not language called for in the standards. You may decide to use it, though.”
• In Unit 3, Lesson 5, Tips for Teachers states, “‘Language plays an enormous role in thinking conceptually about the standard division algorithm. More adults are accustomed to the ‘goes into’ language that is hard to let go. For the problem $$583\div4$$, here is some suggested language, ‘I want to share 5 hundreds, 8 tens, and 3 ones among these 4 sets. There are enough hundreds for each set to get 1 hundred. That leaves 1 hundred that I can’t share. I’ll trade the remaining hundred for 10 tens. That gives me a total of 18 tens. I can give each set 4 tens and have 2 tens left over. Two tens are not enough to go around the 4 sets. I can trade 2 tens for 20 ones and put those with the 3 ones I already had. That makes a total of 23 ones. I can give 5 ones to each of the four sets. That leaves me with 3 ones as a remainder. In all, I gave each group 1 hundred, 4 tens, and 5 ones, with 3 ones left over.’ *Van de Walle, Teaching Student-Centered Mathematics, Grades 3-5, Vol. 2, p. 191).”
• In Unit 7, Lesson 2, Tips for Teachers states, “When saying a number in word form, make sure to only use the word ‘and’ in place of the decimal place and nowhere else. For example, 7.5 is read ‘seven and five tenths.’”
• In Unit 7, Lesson 1, Tips For Teachers states, “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).”

Examples of the materials using precise and accurate terminology and definitions when describing mathematics, and supporting students in using them, include:

• At the beginning of each unit, the Unit Prep provides Vocabulary for the unit. As found in Unit 1, vocabulary includes, but is not limited to, “Ten thousands, millions, hundred millions, hundred thousands, ten millions, variable, etc.”
• In Unit 2, Lesson 7, Criteria for Success, students will, “Understand that a prime number is a whole number that has exactly two factors, 1 and itself.”
• In Unit 3, Lesson 1, Criteria for Success, students will, “Understand that a remainder is the number left over when one number is divided by another.”
• In Unit 4, lesson 3, Criteria for Success, students will, “Understand that perpendicular line segments are line segments that intersect to form right angles.”
• In Unit 4, Lesson 16, Tips For Teachers, “The term ‘trapezoid’ is sometimes defined in two different ways: a quadrilateral with exactly one pair of parallel sides, or quadrilateral with at least one pair of parallel sides. In our curriculum, we choose to use the inclusive definition; a trapezoid as a quadrilateral with at least one pair of parallel sides.” 
• In Unit 5, Lesson 1, Criteria for Success, students will, “Understand that equivalent fractions are fractions that refer to the same whole and are the same size.”