5th Grade - Gateway 2

The instructional materials for Grade 5 partially meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics without losing focus on the major work of each grade. Overall, although word problems are included in the instructional materials, the problems are often routine. Many problems are single-step, and problems that are multi-step are often scaffolded. However, there are ten Problem Solving Lessons designed to help students "isolate and focus on [problem solving] strategies."

In Grade 5 there are several standards that call for application. Standard 5.NF.2 requires students solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators. Standard 5.NF.6 requires that students solve word problems involving multiplication of fractions and mixed numbers. Standard 5.NF.7c requires students solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, and Standard 5.MD.5b requires that students apply the formulas V=l x w x h and V= b x h for rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. Standard 5.MD.5c requires students to recognize volume as additive and find volumes of solid figures composed of two non-overlapping parts, applying this technique to solve real world problems. The instructional materials include some problems that allow students to engage in applications of the mathematics.

• Assessment and Practice Part 1 Unit 5 Lesson 8 aligns to 5.NF.2: “Kim needs 2 1/3 cups of flour to make bread and 3 cups of flour to make dumplings. How much flour does she need altogether?”
• Assessment and Practice Part 1 Unit 5 Lesson 18 Problems 8 and 9 have students multiply a fraction by a whole number. Question 8 states “To make 1 pie, a recipe calls for ¼ of a cup of blueberries. How many cups of blueberries are needed for 8 pies?” Question 9 states “Farah’s exercise routine takes ⅓ of an hour. She exercises 6 days a week. How many hours a week does she exercise?” (5.NF.6).
• In Teacher Resources Part 2 Unit 2, Lessons 29 and 30 have the teacher showing students a ⅓ cup measure, a 1 cup measure and enough counters to fill up the 1 cup measure. The teacher tells students that the small measure is labeled as ⅓ cup and the big measure as 1 cup. The teacher asks the following: “'How many small cupfuls should fill up the big cup?' (3) Ask a volunteer to check that this is the case. Tell students that a recipe calls for cups of flour but you only have the ⅓ cup measure. Teacher asks: 'How many cupfuls do you need?' (6) Have a volunteer write the division question (2 ⅓ = 6)." The sequencing of questions in this problem scaffolds the problem. (5.NF.6)
• In the Teacher Resources Part 2 Unit 2 Lesson 33 Cumulative Review the teacher tells students that some problems that involve division will have a mixed number answer but that the answer needs to be a whole number. The teacher writes on board "Nomi can carry 16 lbs. How many books weighing 1 ½ pounds each can she carry?" There are exercises a-d that are similar for students to practice. (5.NF.7c)
• Assessment and Practice Part 2 Unit 6 Lesson 27 Problem 7 asks students to estimate the answer and then use a calculator to find the actual value. Problem 7a states “The tower of the Aon center in Chicago, IL is a rectangular prism that is 194 ft. wide, 194 ft. long, and 1,123 ft. tall. What is the volume of the tower?” Problem 7b states “The Cheung Kong Center Tower in Hong Kong, China is a rectangular prism 154 ft. wide, 154 ft. long, and 928 ft. tall. What is the volume of the tower?” Problem 7c asks “Which tower has the greatest volume, the Aon Center of the Cheung Kong Center Tower? What is the difference between them?” (5.MD.5b)
• Assessment and Practice Part 2 Unit 6 Lesson 32 Problem 5 states “A skyscraper has three rectangular towers. a) What is the area of the ground floor of the skyscraper? b) What is the area of the top floor of the skyscraper? c) What is the total volume of the skyscraper?” Note – this problem is accompanied by two diagrams. Diagram 1 includes the measures for the ground floor of the skyscraper. Diagram 2 includes the height for each tower. (5.MD.5c)

Word problems can be found in many lessons throughout the instructional materials; however, they are mostly routine, similar to problems previously encountered by students, and/or encourage the use of strategies modeled in the Teacher Resource. As a result, the instructional materials do not present sufficient opportunity for students to engage in non-routine application problems.

• Assessment and Practice Part 2 Unit 2 Lesson 21 Question 7 states: "Mike is making ½ of a recipe for raisin bread. The recipe calls for ⅓ of a cup of raisins. What fraction of a cup of raisins does Mike need?"
• Assessment and Practice Part 2 Unit 2 Lesson 23 Question 4 has students multiplying mixed numbers and mixed numbers by a fraction. Question 4 states: "Luis is making ⅗ of a recipe for mushroom soup. The recipe calls for 3 ½ cups of milk. a) How much milk does he need? Hint: Change 3 ½ to an improper fraction. b) Luis uses 2 cups of milk. Will his recipe work?" The hint scaffolds this problem for students.
• Assessment and Practice Part 2 Unit 2 Lesson 28 Question 8 has students dividing a fraction by a whole number. Question 8 states "Ethan has a scoop that measures a ½ cup. He needs 3 cups of flour. How many spoonfuls of flour does he need?"
• Teacher Resources Part 2 Unit 2 Lesson 29 and 30: "Rosa has two apples. She cuts them each into fourths. How many pieces does she have?"
• Teacher Resources Part 2 Unit 2 Lesson 29 and 30: "Julie baked five muffins that weigh 1 ¾ pounds total. How much does each muffin weigh?"
• Assessment and Practice Part 2 Unit 3 Lesson 13 focuses on Multiplication and Word problems. Students are presented with multiple problems that ask them to identify the larger and smaller quantities in the problem prior to solving the problems. Problem 6, the only word problem, states “Write the equation and replace the correct letter with the given number. If the unknown is not by itself, write the equation that means the same thing. Solve the equation to solve the problem.” Problem 6c states “A snake is 5 times as long as a lizard. The snake is 125 cm long. How long is the lizard?” This is a routine problem using the same strategies used in problems throughout the lesson.

Problem Solving Lessons include word problems that are heavily scaffolded and focused on the use of a particular problem-solving strategy. In Grade 5, none of the Problem Solving Lessons align to application standards.

The instructional materials for Grade 5 partially meet the expectation that the materials balance all three aspects of rigor with the three aspects almost always treated separately within the curriculum including within and during lessons and practice. Overall, many of the lessons focus on procedural skills and fluency with few opportunities for students to apply procedures for themselves. There is a not a balance of the three aspects of rigor within the grade.

• The three aspects of rigor are not pursued with equal intensity in this program.
• Conceptual knowledge and procedural skill and fluency are evident in the instructional materials. There are multiple lessons where conceptual development is the clear focus.
• The instructional materials lack opportunities for students to engage in application and deep problem solving in real world situations.
• There are very few lessons that treat all three aspects together due to the relative weakness in application. However, there are several lessons that include conceptual development leading to procedural practice and fluency.
• There are minimal opportunities for students to engage in cognitively demanding tasks and applications that would call for them to use the math they know to solve problems and integrate their understanding into real-world applications.

The instructional materials reviewed for Grade 5 partially meet expectations that the materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

Materials occasionally prompt students to construct viable arguments or analyze the arguments of others concerning key grade-level mathematics detailed in the content standards; however, there are very few opportunities for students to both construct arguments and analyze the arguments of others together. In the lessons provided in the Teacher Resources Part 1 and 2, examples identified as MP3 are almost always in a whole group discussion, though there are occasional suggestions for students to work in groups. Students rarely have the opportunity to either construct viable arguments or to critique the reasoning of others in a meaningful way because of the heavy scaffolding of the program. For example, in the Teacher Resources Part 2 Unit 5 Lesson 14, the teacher draws a rectangle and labels two sides as 5 inches and 2 feet. The teacher tells the students that she thinks the perimeter of this rectangle is 14 inches. "Is that correct? (no) Have students explain the mistake. (you cannot add feet to inches; you first need to convert the measurements to the same unit) Have students convert both measurements to inches and find the perimeter. (2 ft = 24 in, so (5 in + 24 in) x 2 = perimeter 58 in) Then have students find the perimeter of a rectangle that measures 5 ¾ in by 1 ¼ ft. (1 ¼ ft = 15 in, so the perimeter is 41 ½ in)." This portion of the lesson is labeled with MP3. Although students have to figure out if the answer is correct or not, they aren’t really reaching the full depth of MP3. The teacher talks the students through figuring out the actual perimeter, and the only time that students are constructing an argument themselves is when they are asked to explain the teacher’s mistake. Another example is in Teacher Resources Part 2 Unit 6 Lesson 38. The teacher asks “How many fluid ounces are in 1 gallon? (128) Which fraction of a gallon is 1 cubic inch? (1/231) Which fraction of a gallon is 1 fluid ounce? (1/128) What is larger, 1 cubic inch or 1 fluid ounce? (1 fl oz) How do you know? (231>128, so 1/128>1/231, so 1 fl oz > 1 cubic inch).” These questions lead to understanding but do not address MP3 by having students construct their own arguments and/or critiquing the reasoning of others.

In the Assessment and Practice Books, students are sometimes prompted to construct an argument. For example, in Assessment and Practice Book Part 1 page 105 question 8 asks: “A turtle weighs 4/9 kg and a lizard weighs 5/11 kg. Which animal is heavier? Explain how you know.” Another example is Assessment and Practice Book Part 1 page 18 question 8: “Mona wants to build a model of the number six thousand, five hundred ninety. She has 5 thousands blocks, and 30 tens blocks. Can she build the model? Use diagrams and numbers to support your answer.” Although students are prompted to provide written arguments, often using the word “explain,” students are not provided with formal opportunities to share these written arguments with classmates.

In the Assessment and Practice Books, students are rarely provided opportunities to analyze the arguments of others. When items are included that ask for students to critique the reasoning of others, they are told up front that the student is incorrect. For example, in Assessment and Practice Book 1 page 4 question 9 states “Look at the sequence: 5, 9, 13, 17, 21, … Jake says the rule is : ‘Start at 5 and subtract 4 each time.’ Anika says the rule is: ‘Start at 5 and add 5 each time.’ Ethan says the rule is: ‘Start at 5 and add 4 each time.’ a) Whose rule is correct? b) What mistakes did the others make?”.

The materials reviewed for Grade 5 partially meet the expectation of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

Within lessons, the teacher materials are not always clear about how teachers will engage and support students in constructing viable arguments or critiquing the reasoning of others. Materials identified with the MP3 standard often direct teachers to "choose a student to answer" or "have a volunteer fill in the blank." Questions are provided but often do not encourage students to deeply engage in MP3. In addition, although answers are provided, there are no follow up questions to help redirect students who didn’t understand. Few problems or activities are labeled as MP3.

• Teacher Resources Part 1 Unit 1 Lesson 4 : “Draw the table on the board. Tell students you have a riddle for them: you extended the first sequence for many terms, and you want the students to figure out what the term is in a certain row but in the second sequence. Ask students to explain how they find the answer. (Students need to figure out the rule to get from one sequence to the other, and then to use the rule to determine the number.)” The prompts do not assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others to deeply engage in MP3.
• Teacher Resources Part 1 Unit 5 Lesson 8: In the extension section question 2 the directions state, "Is there a fraction equivalent to ⅜ with an odd denominator? Explain. Answer: No. The denominator is a multiple of 8, so it is even.” There are no directions for the teacher in how to help students engage in constructing viable arguments or analyzing the arguments of others, nor does this help to redirect students who do not understand.
• Teacher Resources Part 2 Unit 7 Lesson 10: In the Acctivity section, students are working with empty regions of Venn diagrams. The directions state: “Have students think for some time to arrive at the conclusion that a shape like that does not exist. Ask them to explain why there cannot be a shape that is both a polygon and a circle. (a polygon has only straight sides, a circle has only one curved side; a polygon has vertices, a circle has no vertices; etc.)” There is no information or directions for the teacher to foster students ability to construct arguments and analyze the arguments of others which does not encourage students to deeply engage in MP3.

Overall, some questions are provided for teachers to assist their students in engaging students in constructing viable arguments and analyzing the arguments of others; however, additional follow-up questions and direct support for teachers is needed.

The materials reviewed for Grade 5 partially meet the expectation for attending to the specialized language of mathematics. Overall, there are several examples of the mathematical language being introduced and appropriately reinforced throughout the unit, but there are times the materials do not attend to the specialized language of mathematics.

Although no glossary is provided in the materials, each unit introduction includes a list of important vocabulary, and each lesson includes a list of vocabulary that will be used in that lesson. The teacher is provided with explanations of the meanings of some words.

• In Teacher Resources Part 1, page A-21 states that “vocabulary words are listed at the beginning of each lesson plan. Make sure students are familiar with the vocabulary words. Make some of the words, such as geometrical terms, part of your spelling lessons.”
• Vocabulary words are listed at the beginning of each lesson plan in the Teacher’s Guide, but definitions, if any, are within the lesson.

While the materials attend to the specialized language of mathematics most of the time, there are instances where this is not the case.

• Often students are not required to provide explanations and justifications, especially in writing, which would allow them to attend to the specialized language of mathematics. For example, in Teacher Resources Part 2 Unit 1 Lesson 1 vocabulary includes the terms array, column, coordinates, ordered pair, and row. Each time, however, that these words are used in the lesson, they are used by the teacher. The student is not required to provide an explanation or justification for their answers that would allow them to use the words in this lesson. After the students do an activity in the lesson it states: "Explain that mathematicians around the world have agreed to give the location of a point by two numbers in parentheses. The column number is always on the left and the row number is always on the right: (column, row). SAY: This means that the numbers in the pair have a specific order, so we call them an ordered pair. Give students several ordered pairs and ask them to identify the corresponding points in an array of dots. Explain that the ordered pair of numbers can also be called the coordinates of the point."
• Many of the discussion prompts provided are guided by the teacher so that the student is merely repeating the teacher's language. This limits student ability to actively use mathematical language.