3rd Grade - Gateway 2

The instructional materials for Into Math Florida Grade 3 meet expectations that the materials develop 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 and provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Throughout the materials, there are sections that emphasize introducing concepts and developing understanding such as: “Build Understanding” and “Spark Your Learning”. Students have the opportunity to independently demonstrate their understanding with the “Check Understanding” and “On My Own” problems at the end of each lesson. For example: 

• Lesson 1.1, Build Understanding 1, students draw a picture to show equal groups and then tell how many groups, with how many in each group, for the problem: “Raj makes 5 robots like this on. How many arms do the 5 robots have?” (3.OA.1.1)
• Lesson 1.2, Spark Your Learning, “Ling is at a fair with her friends Jimmy and Pablo. At a game booth, they each get 4 balls. How many balls will Ling and her friends get? Show more than one way to find the number of balls." (3.OA.1.1)
• Lesson 6.2, Spark Your Learning, “Annie has 20 photographs to put onto pages in a book. She wants to make equal groups. Write questions that can be asked about the scenario. Show equal groups to answer one question." (3.OA.1.1)
• Lesson 13.1, Build Understanding, students compare pictures of flags to determine which are divided into equal parts and which are not. They use this knowledge to draw both types of flags. Then they name equal parts using words like fourths and eighths. (3.NF.1.1)
• Lesson 13.3, On My Own, Question 4, students shade four equal parts of a hexagon, then write the fraction in words and numbers. (3.NF.1.1)
• Lesson 13.6, Practice & HW Journal, Question 7, students are asked, “Each shape represents $$\frac{1}{2}$$ of a whole. Choose how many shapes to put together to make an amount that is great than 1. Draw your shapes.” (3.NF.1.1)
• Lesson 15.3, Practice & HW Journal, Question 8, Math on the Spot, students are asked, “Zach has a piece that is $$\frac{1}{4}$$ of a pie. Max has a piece of pie that is $$\frac{1}{2}$$ of a pie. Max’s piece is smaller than Zach’s piece. Explain how this could happen. Draw a picture to show your answer.” (3.NF.1.3d)

The instructional materials for Into Math Florida Grade 3 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The materials include problems and questions that develop procedural skill and fluency and provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade. Procedural skills and fluencies are primarily found in two areas of the materials. In "On Your Own,” students work through activities to practice procedural skill and fluency; additional fluency practice is found in “More Practice/Homework.”

• Lesson 4.6, Multiply by 9s, On My Own, Problem 5, 9 x 1 = __. Problem 6, 4 x 9 = __. Problem 7, ___ = 9 x 7. (3.OA.3.7)
• Lessons 7.3 and 7.4,  students practice multiplication and division with 2, 3, 4, 5, 6,  8, and 10. Different representations are presented for each operation. (3.OA.3.7)
• Lesson 7.7, students skip count by 2, 3, and 4, and look for patterns. They circle the numbers that are the same in each set. (3.OA.3.7)
• Lesson 9.2 students use mental math to find the sum or difference. For example in, Problem 5, 46 + 24 + ____ and Problem 6, 639 - 425 = ___.
• Lesson 10.2 students estimate and then find the sum of two multi-digit numbers. For example, in Problem 7, students solve 612 + 75; Problem 8, 546 + 56; and Problem 9, 324 + 119. (All problems are presented vertically) (3.OA.3.7)
• Lesson 10.4 students find the difference using place value and regrouping. (3.NBT.1.2)

The instructional materials for Into Math Florida Grade 3 meet expectations that the materials are 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.

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade-level. Students also have opportunities to independently demonstrate the use of mathematics flexibly in a variety of contexts. During Spark Your Learning, Independent Practice, and On My Own, students engage with problems that include real-world context and present opportunities for application. For example:

• Lesson 1.3, On My Own, students use multiplication and division within 100 to solve word problems. For example, “Solar cells on a solar panel collect sunlight which is converted into electricity. Solar cells are arranged on a frame to make a solar panel. Describe one way you could arrange 24 solar cells in an array to make a panel.” (3.OA.1.3)
• Lesson 1.4, Spark Your Learning, students use multiplication and division within 100 to solve word problems in situations. For example, “Andy designed the game board shown (picture provided of a flow chart). The game will have 21 squares. The squares need to be in equal rows. Show two different game board designs that can be made." (3.OA.1.3)
• Lesson 1.5, Exit Ticket, students use multiplication and division within 100 to solve word problems in situations. For example, “Diandre wants to measure the length of a kayak. He has a measuring stick that is 3 feet long.  How can Diandre use the measuring stick and a number line to help him find the total length of the kayak?” (3.OA.1.3)
• Lesson 3.3,On My Own, students solve, “Michael buys 2 packages of hamburger buns. Each package has 6 hamburger buns. How many hamburger buns are there? Show the equal groups. Write an equation for the problem.” (3.OA.1.3)
• Lesson 5.4, Homework, Problem 9, students solve two-step word problems using the four operations. For example, “Ava’s class bought 6 packages of balloons for a school celebration. Each package had 30 balloons. If 17 balloons were left over, how many balloons were used for the party?” (3.OA.4.8)
• Lesson 6.2, Spark Your Learning, students determine how 20 photos might be arranged into a photo album using equal groups. (3.OA.1.3)
• Lesson 8.4, On My Own, students write two equations with letters representing unknown to solve the problems.  An example is, “Jamie’s plant grows 3 inches each week for three weeks. During the fourth week, it growth 5 inches. How much does Jamie’s plant grow over the four weeks?” (MAFS.3.OA.4.8)
• Lesson 10.6, On My Own, students solve two-step word problems using the four operations. For example, “Write a two-step word problem with an unknown number. Write equations to model the problem. Then solve.” (3.OA.4.8)

Each Unit has a Performance Task involving real-world applications of the mathematics from the unit. For example, the Unit 3 Performance task has students follow one child, a baseball card collector, throughout his day. Students estimate the number of baseball cards he has at the end of the day (3.NBT.1.1), tell the time of his various stops throughout the day (3.MD.1.1), determine the perimeter of his baseball card display (3.MD.4.8), and how many coins he has but not counting money (3.OA.1.3).

The instructional materials for Into Math Florida Grade 3 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. In general, two, or all three, of the aspects are interwoven throughout each module. The module planning pages include a  progression diagram showing the first few lessons focused on understanding and connecting concepts and skills. The last lessons focus on applying and practicing.

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

• Modules 4, 7, 9 and 10, students develop procedural skill and fluency with finding products, working with related facts, and division. Students also implement estimation and mental math to support the addition and subtraction for the grade level. (3.OA.3.7, 3.NBT.1.2)
• Lesson 8.4, On My Own, students engage in application in Problem 7, “Write a two-step word problem that can be solved using two equations with different operations.” (MAFS.3.OA.4.8)
• Lesson 16.1 builds conceptual understanding of equivalent fractions. Students draw visual models and use number lines to show fraction equivalence. (3.NF.1.3b)

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

• Lesson 7.6 attends to procedural skill and application related to multiplication and division with seven and nine. For example, “The 28 students in Van’s class are on a field trip to a cave. They are divided into groups of 7 for a tour of the cave. How many groups are there?” (3.OA.2.5)
• Lesson 13.6, Spark Your Learning, attends to conceptual understanding and application. For example, “Emilio cuts his pizzas into fourths. If he has 9 slices, show the different amounts of pizza that Emilio could sell. Name each fraction and show.” (3.NF.1.1 and 3.NF.1.2)
• Unit 4, Performance Task, attends to conceptual understanding and application. Students create number lines to visually see equal parts and then use them to solve real-world problems. For example,“Yoshi makes a large sandwich, he cuts the sandwich into 8 equal parts. He wants to put $$\frac{5}{8}$$ of the sandwich on a plate. How many pieces of the sandwich does he need? Draw a number line to solve the problem. Then write your answer.”
  

The instructional materials reviewed for Into Math Florida Grade 3 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Student materials consistently prompt students to construct viable arguments and analyze the arguments of others. A common strategy in these materials is Turn and Talk with a partner about the related task. Often these Turn and Talks require students to construct viable arguments and analyze the arguments of others. In addition, students justify their reasoning in practice problems, especially those labeled “Critique Reasoning.”

•  In Lesson 5.1, Problem 4, “Pam says that she can write the product of 8 x 60 as the sum of two products. Is she correct? Explain.”
• In Lesson 7.1, On My Own, students critique the reasoning of two students who have different ways to solve the same problem, one using division and the other multiplication.
• In Lesson 10.6, On My Own, Problem 3, students analyze the work of others. For example, there is a box labeled “Ed’s Work” with two equations in it. “Ed has 8 boxes with 7 rocks in each box. Then he finds 9 more rocks. Ed writes these equations to find how many rocks he has now. Is Ed’s work correct? Explain.”
• Lesson 11.4, Turn and Talk, “When rectangles have the same area, how do you know which will have the greatest perimeter and the least perimeter?"
• Lesson 13.2, Turn and Talk, “Compare the fractions in Task 1 and 2. How are they alike? How are they different?"
• Lesson 14.1, Turn and Talk, “What fraction of the whole playground is in each hamster’s play area? How do you know?”
• In Lesson 16.2, Spark your Learning, “Thea is a landscaper. According to her design, $$\frac{2}{3}$$ of the garden should contain red roses. She planted red roses in $$\frac{4}{6}$$ of her new garden, not $$\frac{2}{3}$$. Has Thea made a mistake? Show a way to solve the problem.”
• In Lesson 16.2, Build Understanding, “Peter’s barn is sectioned off into 8 horse stalls. Peter uses $$\frac{6}{8}$$ of the stalls for his jumping horses. Peter uses the 2 stalls on the far right side of the barn for storage. How do the fractions compare for the area of the barn for jumping horses in each of Peter’s set-ups. Explain.”
  

The instructional materials reviewed for Into Math Florida Grade 3 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Many of the lesson tasks are designed for students to collaborate, with teacher prompts to promote explaining their reasoning to each other. Independent problems provided throughout the lessons also have teacher guidance to assist teachers in engaging students.

• The Teacher Edition provides Guided Student Discussion with guiding questions for teachers to create opportunities for students to engage in mathematical discourse. For example, in Lesson 4.7, Step it Out, Sample Guided Discussion, students are asked, “How can you tell if a product will be even? How can you represent the number 4 as the sum of two equation addends? So, if you multiply 7 by (2 + 2), what are the two smaller products? What statements can you write about whether the product of two odd numbers is even or odd? How can you prove why your statement makes sense using equal groups?”
• Critique, Correct, and Clarify is a strategy used to assist students in constructing viable arguments. For example, in Lesson 2.2, On My Own, Problem 7, students analyze a statement made by a fictitious student. Teachers are told to “Point out to students that in Problem 7, Katy’s statement about a gap when measuring the area of a figure may or may not be correct. Encourage students to describe why the statement is or is not correct and to review explanations with a partner. Students should refine their responses after their discussions with a partner.” In Lesson 9.6, On My Own, Problem 6, students analyze two ways to estimate the difference of 524 - 365, and tell which estimate will be closer to the actual difference. Teachers are told to “Encourage students to describe why they think one estimate is closer to the actual difference and to review explanations with a partner. Students should refine their responses after their discussions with a partner.”
• The Teacher Edition includes Turn and Talk in the margin notes to prompt student engagement. For example, in Lesson 1.3, “Select students who used various strategies and have them share how they solved the problem with the class. Encourage students to ask questions of their classmates. As a class, choose a number of rows of chairs. Make a visual model of the rows.”
• In Lesson 14.2, Connect Math Ideas, Reasoning, and Language states, “Remind students that they have seen that equal parts have equal areas. Have students describe and give examples in their own words of what area is. Have them share their work and discuss how their descriptions compare and contrast.”

The instructional materials reviewed for Into Math Florida Grade 3 meet expectations for explicitly attending to the specialized language of mathematics. The materials provide explicit instruction on communicating mathematical thinking with words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. Examples found throughout the materials include: 

• At the beginning of each module, Key Academic Vocabulary is highlighted for the teacher.  The sections include both Prior Learning - Review Vocabulary, and Current Development - New Vocabulary.  Definitions are given for each vocabulary word.
• Within the lessons, new vocabulary is introduced in highlighted sections called Connect to Vocabulary.  For example, in Lesson 13.1, "A whole is all of the parts that make up one shape or group. If all of the parts of a whole are the same size, then the whole is divided into equal parts.” In Lesson 10.2, “To regroup is to exchange amounts of equal value to rename a number. Examples: 17 ones is one ten and seven ones. 13 tens is one hundred and 3 tens.”
• In the Module planning pages, a Linguistic Note on the Language Development page provides teachers with possible misconceptions relating to academic language. For example, In Module 2, “Listen for students who may not be comprehending instructions that use formal math language. For example, consider the following instructions that use formal math language. What is the area of the figure? Draw a vertical or horizontal line to break apart the figure into smaller rectangles. Complete the equations. After reading the formal language to students, restate the instructions using everyday language: Draw a line that goes up and down. Draw a line that goes across from side to side. What does a rectangle look like? How can you make small rectangles?”
• In Sharpen Skills in the lesson planning pages, some lessons include Vocabulary Review activities. For example, in Lesson 10.1, “Objective: Students complete graphic organizers for the review terms sum and expanded form.” “Materials: Word Descriptions graphic organizer,” “Have students form small groups and complete the graphic organizer shown for the term sum. Students can take turns stating an example of a sum and having the others in the group decide whether it is an example or a non-example. Repeat using a new graphic organizer for the term expanded form.”
• Guide Student Discussion provides prompts related to understanding vocabulary such as in Module 1, “Listen for student who correctly use review vocabulary as part of their discourse.  Students should be familiar with the terms sum, addend, and equal groups.  Ask students what they mean if they use those terms.” “How could you use an array to represent each total?” “How would you count the objects in the array?” “Is there more than one way to use equal groups to count the objects in an array of 2 rows with 6 objects in each row? Explain.”
• Student pages include Connect to Vocabulary boxes that define content vocabulary. In Lesson 4.1, “The Identity Property of Multiplication states that the product of any number and 1 is that number.”
• Vocabulary is highlighted and italicized within each lesson in the materials.
• There is a vocabulary review at the end of each Module. Students complete a fill-in-the-blank with definitions or examples, create graphic organizers to help make sense of terms, or the teacher is prompted to make an Anchor Chart where students define terms with words and pictures, trying to make connections among concepts.