3rd Grade - Gateway 2

​The instructional materials for enVision Florida Mathematics Grade 3 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The structure of the lessons includes several opportunities to develop conceptual understanding.

• In the Teacher Edition, every Topic begins with Math Background: Rigor, where conceptual understanding for the topic is outlined.
• Lessons are introduced with a video, Visual Learning Animation Plus, at PearsonRealize.com; these often build conceptual understanding.
• Links within the digital program to outside resources, such as Virtual Nerd, include videos for students that introduce concepts.
• In the student practice problems, the section Do You Understand reviews conceptual understanding.

Materials include problems and questions that develop conceptual understanding throughout the grade level and provide opportunities for students to demonstrate conceptual understanding independently throughout the grade.

• In Lesson 12-1, the Visual Learning Animation Plus states, “How Can You Name the Equal Parts of a Whole?” The scenario begins by having students divide sections of green wholes into four equal parts. The visual shows correct ways, as students fold their paper, along with incorrect ways, and the animation discusses ¼ as a unit fraction that represents one of the equal parts.
• The Topic 1 Overview, Conceptual Understanding states, “Throughout Topic 1, students build their conceptual understanding of how multiplication and division relate to equal-group situations. They come to understand that equal-group situations can be represented using multiplication or division depending on what information is known and what is unknown. Bar diagrams help students understand and explain how the numbers are related.” In Lesson 1-3, students use counters to build groups and evaluate multiplication problems. Students draw arrays to show equal groups.
• The Topic 6 Overview, Conceptual Understanding states, “Understand area as covering with unit squares.” In Lessons 6-1, 6-2, and 6-3, students count unit squares, both nonstandard and standard, to find the area of figures. This explicit focus on area as covering with unit squares helps students develop conceptual understanding of area.
• The Lesson 13-7 Lesson Overview, Conceptual Understanding states, “Students use the knowledge they have gained about fraction strips, number lines, and equivalencies to find fraction names for whole numbers.” By emphasizing that whole numbers have many fractions names, the lesson reinforces the understanding of fractions as numbers.

​The instructional materials for enVision Florida Mathematics Grade 3 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

Examples of the the instructional materials developing procedural skills and fluencies throughout the grade level include:

• Procedural skills and fluencies integrate with conceptual understanding and the work students completed with operations from prior grades. Opportunities to practice procedural skills are found throughout practice problem sets that follow the units and include opportunities to use fluencies in the context of solving problems.
• The Teacher Edition Program Overview articulates, “Steps to Fluency Success.” The six steps are: Step 1: Fluency Development with Understanding, Step 2: Ongoing Assessment of Fluency Subskills, Step 3: Fluency Intervention, Step 4: Practice on Fluency Subskills, Step 5: Fluency Maintenance, and Step 6: Summative Fluency Assessment. Fluency Expectations for Grades K-5 are also listed. The Teacher Edition Topic Overview explains the six steps and foundations for fluency. In each Topic Overview, Math Background: Rigor, there is a section explaining how the material builds Procedural Skill and Fluency. The Topic 5 Overview, Procedural Skill and Fluency states, “Fluency with multiplication and division within 100 is an expectation in this topic.” Students are provided opportunities to interpret multiplication tables and use other strategies to multiply and divide.
• Within each lesson, the Visual Learning Bridge integrates conceptual understanding with procedural skills. Additional Fluency and Practice pages are in the Teacher Edition and Ancillary Books as well as online with the Practice Buddy Additional Practice. The online component also contains a game center where students continue to develop procedural skills and fluencies. Each topic ends with Fluency Practice/Assessment Worksheets.

The instructional materials provide opportunities for students to  demonstrate procedural skill and fluency independently throughout the grade level.

• In Lesson 5-2, Independent Practice, students find the missing factors and products in a table.
• In Lesson 5-3, Independent Practice, students use strategies to find the products.
• In Lesson 16-2, Visual Learning Bridge, Convince Me!, Guided Practice, and Independent Practice sections of the lesson students use multiplication within 100 to find the perimeter of common shapes.

The instructional materials provide regular opportunities for students to attend to the Standard 3.NBT.1.2, adding and subtracting within 1,000 and to the Standard 3.OA.3.7, multiplying and dividing within 100.

​The instructional materials for enVision Florida Mathematics Grade 3 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.

Work with applications of mathematics occurs throughout the materials. In each Topic Overview, Math Background: Rigor explains how the materials utilize applications. For example, the Topic 2 Overview, Math Background: Rigor states, “Students solve a variety of real-world problems involving equal groups and arrays. These situations allow students to apply multiplication facts to their understanding of multiplication.”

Following the Topic Overview, the Topic Opener includes an enVision STEM Project where application activities are provided and can be revisited throughout the topic. In each topic, Pick A Project allows students to explore areas of interests and to complete projects that apply the mathematics of the topic. Every other topic contains 3-Act Math where students engage in mathematical modeling.

At the end of each topic, the Performance Task provides opportunities for students to apply the content of the topic. Additional application tasks are in Additional Practice pages in the Teacher Edition, Ancillary Books, and online.

Examples of opportunities for students to engage in routine and non-routine application of mathematical skills independently and to demonstrate the use of mathematics flexibly in a variety of contexts include:

• In Topic 1, 3-Act Math, students solve multiplication and division problems in the context of a story problem. Students complete the three acts to solve the main question, “How many packs of pencils will it take to fill 3 cups?” In Act 1: THE HOOK, students watch an informational video, brainstorm questions, and predict reasonable answers to the main question. In Act 2: THE MODEL, students identify and reveal information as well as develop a model. In Act 3: THE SOLUTION, students reveal an answer and reflect to validate conclusions, revise their models, discuss math practices, and revisit brainstorming.
• In Lesson 2-6 Problem Solving Performance Task, students solve multiplication and division problems in the context of a story problem. “David and Jon are placing coffee orders for their friends. David orders 10 large cups of coffee. Jon orders 4 fewer large cups than David. Jon pays for his orders with a $50 dollar bill. Jon wants to know how much he spent on coffee.” Students are provided with information on the cost of different sizes of coffee cups. 
• In Lesson 5-4 Problem Solving, students solve multiplication and division problems in the context of a story problem. “Jodie has 24 flowers in her garden. She wants to give an equal number of flowers to 4 families in her neighborhood. How many flowers will each family get? Complete the bar diagram and write an equation to help solve this problem."
• In the Topic 11 Performance Task, students solve two-step word problems using the four operations in the context of a story problem. “Mrs. Radner and Mr. Yu teach filmmaking at a summer camp. The students work in crews to make movies. The summer ends with the crew and actors watching all the movies.” Class detail information is provided, along with two tables (Film Types and Film Lengths). Students use this information to answer 5 problems.
• In Lesson 11-4 Problem Solving Performance Task, students solve two-step word problems using the four operations in the context of a story problem. “A Grade 3 class is going to buy buttons like the ones shown. Each package costs $8. Each package is 40 cm long. They need to know if $50 is enough money to buy 200 buttons.”

​The instructional materials for enVision Florida Mathematics Grade 3 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The instructional materials address specific aspects of rigor, and the materials integrate aspects of rigor.

Each lesson contains opportunities for students to build conceptual understanding, procedural skills, and fluency, and to apply their learning in real-world problems. During Solve and Share and Guided Practice, students explore alternative solution pathways to master procedural fluency and develop conceptual understanding. During Independent Practice, students apply the content in real world applications and use procedural skills and/or conceptual understanding to solve problems with multiple solutions and explain/compare their solutions.

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

• In Lesson 3-2, students develop conceptual understanding by using visual models such as pictures, drawings, and counters to show multiplication.
• In Lesson 2-2, students practice the procedural skill of multiplying by 1 and 0 using patterns to gain fluency in the products for multiplication with 0 and 1. “Kira has 8 plates with 1 orange on each plate. How many oranges does Kira have?”
• In Lesson 11-1, students apply knowledge of addition and subtraction by using bar diagrams, equations, and information from a table to solve a two-step problem and determine how they relate. “Write equations to find how many more tickets were sold for the roller coaster on Saturday than for the swings on both days combined. Use letters to represent the unknown quantities.”

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

• In Lesson 13-8, Independent Practice, students develop conceptual understanding of fractions through drawing diagrams and shading the fraction, then applying this information to solve, “Reyna has a blue ribbon that is 1 yard long and a red ribbon that is 2 yards long. She uses $$\frac{2}{c}$$ of the red ribbon and $$\frac{2}{4}$$ of the blue ribbon. Conjecture: Reyna uses the same amount of red and blue ribbon.”

​The instructional materials reviewed for enVision Florida Mathematics Grade 3 meet expectations that the Standards for Mathematical Practice (MPs) are identified and used to enrich mathematics content within and throughout the grade level.

Examples of the MPs being identified at the topic level include:

• In Topic 1, the Overview identifies MP.1.1. “Students make sense of problems and use counters, bar diagrams, drawings, or equations to represent their work.”
• In Topic 4, the Overview identifies MP.7.1. “Students analyze the relationship between multiplication and division by solving division facts using related multiplication facts.”

The MPs are used to enrich the mathematical content and are not treated separately. MPs are highlighted and discussed throughout the lesson narratives, and along with the lessons, the MPs are evident in the the 3-Act Math Tasks that are included in every other chapter. The MPs are listed in the student materials, and the Math Practice Handbook is available online for teachers to make available to students.

• In Lesson 15-1 Convince Me, MP.1.1 is identified. “Students draw a quadrilateral that is an example of one of the shapes listed in Box B and then name the shape. They also draw a quadrilateral that is not an example of a shape listed in Box B. Check students’ drawings to make sure that they understand the attributes of different quadrilaterals.”
• In Lesson 4-4, problem 19, MP.2.1 is identified. “What other equations are in the same fact family as $$18\div9$$ = 2? After students have listed the facts in the fact family, have them explain why the facts are a fact family.”
• In Lesson 7-5, problem 7, MP.6.1 is identified. “Use precise math language and symbols. Students accurately explain how Marta can buy 12 sketches for $50 or less. Ask students if they could have solved the problem another way.”
• The Topic 13, 3-Act Math Task identifies MP.4.1 as the primary Math Practice but connects to other MPs: “As students carryout mathematical modeling, they will also engage in sense-making (MP.1.1), abstract quantitative reasoning (MP.2.1), and mathematical communication and argumentation (MP.3.1). In testing and validating their models, students look for patterns in the structure of their models." (MP.7.1, MP.8.1)

The MPs are identified within a lesson in the Lesson Overview, and the lesson narratives highlight when a MP is particularly important for a concept or when a task may exemplify the identified Practice. The lessons that end each Topic specifically focus on at least one MP. For example:

• In Lesson 2-6, Problem Solving, MP.4.1 is identified. “Students use number lines as another way to represent the problem. On the top number line, students can show the addition of 2 + 6, and on the bottom number line, they can show multiplication by skip counting by 2's eight times.”
• In Lesson 6-6, MP.7.1 is identified. “Students find the relationship between the area of a decomposed shape and the area of the irregular shape.”
• In Lesson 3-7, Problem Solving: Repeated Reasoning Guided Practice, MP.8.1 is identified. “Listen and look for these behaviors as evidence that students are exhibiting proficiency with this practice: Notice and describe when certain calculations or steps in a procedure are repeated, generalize from examples or repeated observations, recognize and understand appropriate shortcuts, and evaluate the reasonableness of intermediate result.

​​The instructional materials reviewed for enVision Florida Mathematics Grade 3 meet expectations for prompting 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. The Solve and Share Activities, Visual Learning Bridge Problems, Problem Sets, 3-Act Math, Problem Solving: Critique Reasoning problems, and Assessments provide opportunities throughout the year for students to construct viable arguments and analyze the arguments of others.

Examples of the instructional materials supporting students to analyze the arguments of others include:

• In Lesson 1-6, Problem Solving, Question 10: "Critique Reasoning: Kerry says she can use a tens rod to represent the array. Do you agree? Explain.” Students have to use their understanding of arrays to critique Kerry’s reasoning.
• Lesson 2-5, Problem Solving, Question 27: “Abdi says that 9 x 6 is less than 10 x 4 because 9 is less than 10. Do you agree with Abdi’s reasoning? Explain why or why not.”
• In Topic 4, Topic Assessment, Question 11: “Mandy is trying to find 6 0. She says the answer is 6 because 6 x 0 = 6. Is Mandy correct? Explain.”
• In Lesson 8-3, Problem Solving, Question 16: “Critique Reasoning: Bill added 438 + 107. He recorded his reasoning below. Critique Bill’s reasoning. Are there any errors? If so, explain the errors. Find 438 + 107. I’ll think of 7 as 2 and 5. 438 +2 = 440, 440 + 7=447, 447 + 100= 547, so, 438 + 107 is 547.”

Examples of the instructional materials prompting students to construct viable arguments include:

• In Lesson 4-7, Convince Me, students construct arguments to explain the relationship between multiplication and division equations, finding that both belong to the same fact family. “Why can both $$28\div7$$ = ? and ? x 7 = 28 be used to solve the problem above?” Students can use the argument of the relationship between multiplication and division to solve this problem.
• In Lesson 5-6, Problem Solving, Question 9, “Construct Arguments: Compare the costs of buying the $4 packages to the $6 packages. Which package type costs less if Trina wants to buy 24 necklaces? Explain how to solve without computing.”
• In Lesson 9-7, students use the digits 0, 1, 2, 3, 4, and 5 once to make two 3-digit addends with the greatest sum. “How do you know you have made the greatest sum?”

​​The instructional materials reviewed for enVision Florida Mathematics 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.

Teacher materials assist teachers in engaging students in constructing viable arguments and/or analyzing the arguments of others throughout the program. Many of the activities are designed for students to work with partners or small groups where they collaborate and explain their reasoning to each other.

• In Lesson 1-3, Visual Learning Bridge Classroom Conversation, Part B: “You can use addition or skip counting to find the total. The teacher asks students, “Why do you add 5 four times? How would the addition change if the array had another row?”
• In Lesson 2-5, Critique Reasoning prompts the teacher to ask questions related to the mistakes a student makes when estimating a product.
• In Topic 3, 3-Act Math, Critique Reasoning states, “Have students share their solution methods." Let students ask questions about others' solutions.
• In Lesson 9-7, Solve and Share prompts teachers with the following questions to assist students in constructing viable arguments for their work: “What are you asked to do?; What information will you use?; How might you use place value to help solve this problem?; “What is the value of the greatest place value in the boxes?; What is the sum of the two greatest numbers in the group of six numbers?”
• In Lesson 10-3, Construct Arguments (in the margin), students answer questions and construct arguments about why 4 x 20 could be grouped as 4 x (2 x 10).
• In Lesson 11-4, Visual Learning Bridge Classroom Conversation, teachers have the following questions to assist students in analyzing the arguments of others: “What is the main question you need to answer to check Danielle’s reasoning?; What is the hidden question?; What strategy did Danelle use?; What calculations of Danielle’s do you need to check?”

​The instructional materials reviewed for enVision Florida Mathematics Grade 3 meet expectations for explicitly attending to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them.

• The Grade 3 Glossary is located in the Teacher Edition Program Overview, and the Glossary is also present at the back of Volume 1 of the Student Edition.
• Lesson-specific vocabulary can be found at the beginning of each lesson under the Lesson Overview, with words highlighted in yellow used within the lesson, and a vocabulary review is provided at the end of each topic.
• There is a bilingual animated glossary available online that uses motion and sound to build understanding of math vocabulary and an online vocabulary game in the game center.
• Both the topic and the lesson narratives contain specific guidance for the teacher to support students to communicate mathematically. Within the lesson narratives, new terms are highlighted in yellow and explained as related to the context of the material.
• The Teacher Edition Program Overview, Building Mathematical Literacy, outlines the many ways the materials address mathematics vocabulary, including: My Word Cards, Vocabulary Activities at the Beginning of Each Topic, Vocabulary Reteach to Build Understanding, Vocabulary and Writing in Lessons (where new words introduced in a lesson are highlighted in yellow in the Visual Learning Bridge, and lesson practice includes questions to reinforce understanding of the vocabulary used), Vocabulary Review at the back of each topic, an Animated Glossary where students can hear the word and the definition, and Vocabulary Games Online. There is also Build Mathematical Literacy within each Topic Overview that outlines support for English Language Learners, Mathematics Vocabulary, and Math and Reading within the topic.
• In Topic Planner, there is a vocabulary column that lists the words addressed within each lesson in the topic. For example, in the Teacher Edition, Lesson 13-1, the following words are listed: equivalent fractions. These same words are listed in the Lesson Overview.
• Lesson 2-3 introduces the Identity (One) Property of Multiplication and Zero Property of Multiplication. Within the Visual Learning Bridge, students work with equal groups. The definition of the Identity (One) Property of Multiplication is developed and applied as students place 8 groups with 1 in each group using correct language. For example, “The Identity (One) Property of Multiplication, when you multiply a number by 1, the product is that number, 1 plate with 8 oranges also equals 8 oranges.” Within the Convince Me! Activity, students use counters to show 7 x 1 using the Identity Property of Multiplication. The Classroom Conversation provides further practice and discussion questions for the teacher that will solidify the concept of the Identity (One) Property of Multiplication. “How could you use counters to show the number of oranges on this plate? What property lets you know that 8 x 1 = 1 x 8? Can you show 0 x 4 with counters? What property lets you know that 0 x 4 = 0?”
• In Lesson 6-1, students interpret square units by using estimation. Students use the context to build proper mathematical vocabulary.
• Lesson 9-2 introduces regrouping to the students. The Visual Learning Bridge includes definitions and models/diagrams using this new vocabulary. In Guided Practice, students are provided questions within the context of the lesson to answer using vocabulary. For example, question 1: “When you add 3-digit numbers, how do you know if you need to regroup?” A sample answer is provided to support teachers using precise vocabulary language with students.

No examples of incorrect use of vocabulary, symbols, or numbers were found within the materials.