3rd Grade - Gateway 2

The instructional materials reviewed for Ready Classroom Mathematics Grade 3 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. 

Lessons are designed to support students to explore and develop conceptual understanding of grade-level mathematics. For example, students develop conceptual understanding:

• In Student Worktext, Lesson 4, Session 1, Explore, students interpret products of whole numbers as the total number of objects to develop conceptual understanding related to the 3.OA.1. A variety of problems are provided for students to solve including modeling with equal groups, arrays, and using an equation.  Model It, Problem 4 states, “You can model multiplication with equal groups. Circle the tennis balls to show 3 equal groups of 4 tennis balls.” Problem 5 shows the equation 3 x 4 = 12 and describes what each part of the equation represents. Problem 6 has students model 3 x 4 = 12 using arrays.  
• In Student Worktext, Lesson 10, Session 2, Develop, students develop an understanding of division models and interpret whole-number quotients of whole numbers (3.OA.2).  Model It: Equal Groups, Problem 1 states, “Marc has 24 oranges to put in bags. He decides to put 6 oranges in each bag. A. Draw a model to show how many bags he has. B. Write the division equation for your model. C. Use words to describe the total number of oranges, number in each group, and the number of groups.”
• In Lesson 33, Session 1, Explore, Connect It, students explain fraction units of a shaded part. Teachers give students visual models such as grid paper to help students develop a conceptual understanding of partitioning shapes into equal parts.  In Session 2, Connect It, students fold a paper rectangle to divide. Students use a grid paper to answer the question, “Brett folded a piece of paper three times as shown. He then colored $$\frac {1}{4}$$ of the total area red. How could he have colored his paper? Explain how you know your way is right.” In Session 3, Develop, Reteach, students use unit tiles to form rectangles and identify fractional areas. (3.G.2)

In the Student Worktext and during Interactive Practice, students have opportunities to independently demonstrate conceptual understanding. For example:

• In Lesson 9, Session 2, Develop, Practice Multiplying with Tens, Problem 6 states, “Write the multiplication equation that the base ten model shows.” Students are shown six groups of base ten rods with 30 in each group. (3.NBT.3)
• In Lesson 20, Session 2, Develop, students independently complete eight problems identifying fractions of a shaded part, shading parts of a whole, and naming the whole when given a part. For example, Problem 8 shows a right triangle and asks students to complete the following task, “This is $$\frac {1}{4}$$ of a rectangle. Draw the rectangle. Show the parts. Then shade $$\frac {2}{4}$$ of your rectangle.” (3.NF.1)
• In Lesson 32, Student Worktext, Session 4, Develop, Practice Finding Same Perimeter with Different Area, Problem 5 states, “Draw a rectangle that has the same perimeter as Rectangle A and a different area than Rectangles A, B, and C. Write the length, width, and area of the rectangle.” (3.MD.8)
• In Interactive Practice, Multiply with 3, 4, 6, students use arrays and matching to develop understanding of multiply by 3, 4, and 6. (3.OA.1)
• In Interactive Practice, Understand the Meaning of Division, students use arrays and the relationship with multiplication to understand division. (3.OA.2)

The instructional materials reviewed for Ready Classroom Mathematics Grade 3 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional materials include problems and questions, interactive games, and math center activities that develop procedural skill and fluency throughout the grade. For example, in Classroom Resources:

• In Lessons  2 and 3, Fluency and Skills Practice, students add and subtract three-digit numbers. Specific problems include but are not limited to: “102 + 107, 317 + 283, 970 - 625, and 882-511.” (3.NBT.2) Students develop procedural and fluency skills with adding and subtracting using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
• In Lesson 27, Session 1, Explore, Prepare for Working with Time, Problem 3 states, “Brie starts watching a movie at the time shown on the clock. What time does the clock show?” (3.MD.1)

The instructional materials include Learning Games, interactive games to help build procedural skill and fluency, available in both English and Spanish, that can be accessed through i-Ready Reports. For example, Grade 3 students can play the game “Cupcake” which allows them to strengthen computational skills in the areas of addition and subtraction. (3.NBT.2) In this game, students are tasked with running a cupcake delivery service and must add and subtract to purchase ingredients and run their business. 

3.OA.7 (Fluently multiply and divide within 100) is addressed in several lessons, including:

• In Lesson 12, Fluency and Skills Practice consists of 20 mixed problems where students fill in the missing numbers in the multiplication or division problems. Empty boxes range from the product to the multiples. Some specific problems include but are not limited to: “5 x 7 = __”, “____ ÷ 5 = 7”, and “81÷  __ = 9.”
• In Lesson 15, Interactive Tutorial includes a twenty minute tutorial to “Add and multiply to find area.” Students develop fluency in multiplying within 100 (3.OA.7), as they connect finding the area of a rectangle by covering the rectangle with rows and columns of square units, with multiplying its side lengths to find the area. Students write and solve multiplication problems to find the area. For example, “3 x 5 = 15.” Students also break rectangles into smaller rectangles and break apart the numbers to complete the multiplication. For example, Instruction, Part 10, Gabes Blanket presents a rectangle with the dimensions “9 inches x 6 inches.” The problem 9 inches  x 6 inches can be broken into two equations with friendly numbers because 5 + 4 = 9. Students find that 5 inches x 6 inches = 30 inches$$^2$$ and 4 inches x 6 inches = 24 inches$$^2$$. Students then determine that 30 inches$$^2$$ + 24 inches$$^2$$ = 54 inches$$^2$$.

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. Within each lesson, there are Fluency and Skills Practice pages that children complete on their own. In addition, there are Learning Games and Math Center Activities that engage students with fluency practice. Examples of when students get opportunities to independently demonstrate procedural skill and fluency include:

• In Lesson 9, Fluency and Skills Practice, Multiplying with Tens, Problem 3 states, “7 x 20 = ___.” Problem 5states, “50 x 4 = ___.” 
• In Lesson 32, Fluency and Skills Practices states, “A triangle has sides that are all the same length. If the perimeter of the triangle is 27 inches, what is the length of one side?” (3.MD.8, 3.OA.7)
• Math Center Activities in Ready Classroom Mathematics Grade 3 provides students with an opportunity to demonstrate procedural skill and fluency. In Lesson 7, Multiply with 7, 8, and 9, students develop procedural skills in multiplying within 100 (3.OA.7) by playing a structured game called Multiplication Race with a partner. The game requires students to use factor cards with the numbers 7, 8, and 9, multiplier cards with the numbers 1-10, and a game board to find products and move around the board. Their partner works to check the answer of their peer and if they are correct, they can move forward either one or two spaces based on the selected factor card. Students win the game by being the first one to get around the board.

The instructional materials reviewed for Ready Classroom Mathematics 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. 

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

• In Lesson 15, Session 4, Refine, Apply It, Problem 4 states, “Mr. Frank is putting tile on a bathroom wall above the tub. The model shows the length and width of the wall. How many square feet of tile does he need to cover the wall?” There is a diagram of a rectangular room that is 7 ft X 6 ft.
• In Lesson 18, Session 4, Refine, Try It states, “A zoo names an elephant Tiny. On Saturday, Tiny ate 152 pounds of food. On Sunday, he ate 12 more pounds of food than he did on Saturday. How many pounds of food did Tiny eat that weekend? Estimate to check your answer.” 
• In Lesson 27, Session 3, Develop, Try It states, “Jenna gets home from school at 3:30 pm. She does math homework for 10 minutes. Next she does science homework for 15 minutes. Then she practices the piano for 22 minutes. What time does Jenna finish?”

The instructional materials include multiple opportunities for students to engage in non- routine application of mathematical skills and knowledge of the grade level. 

For example, in Classroom Resources:

• In Unit 2, Math in Action states, “Brandi is planning how to set up seats for a play. My Notes: Use between 80 and 100 seats. Make 2 sections. The number of seats in each section can be the same or different. Use equal rows of seats in a section.” Students must also solve, “Help Brandi set up the chairs. Decide the number of chairs to use. Tell how many sets to put in a section. Tell the number of rows and the number of seats in each row.” 
• In Unit 4, Unit Review, Performance Task states, “The owner of the neighborhood pizzeria, Itsa Pizza, would like you to draw diagrams to show different combinations of toppings on 6 pizzas. Each diagram will show a rectangular pizza cut into 8 equal-sized pieces. She wants each pizza to be completely covered with toppings with no overlaps.” Students are provided with the fraction of each of the six pizza types, for example, “The Green Hula includes $$\frac{3}{4}$$ onion, $$\frac{3}{3}$$ pineaplle, $$\frac{1}{4}$$ broccoli.” Students record their work on grid paper using diagrams. 
• In Unit 5 Math in Action, Session 2, Persevere on Your Own states, “Max plans to make tomato soup. His recipe makes 24 liters of soup. He will freeze the soup in containers. Then he’ll have plenty of soup snacks ready to go. Max wants to buy some 1-liter containers for the soup. He can buy different packages of 1-liter containers: Package of 4 containers; Package of 5 containers; Package of 6 containers. What packages should Max buy?” Students need to “Tell how many containers Max needs. Tell which packages Max should buy. Tell how many of each package he should buy. Show why your solution gives the exact numbers of containers Max needs.”

The instructional materials reviewed for Ready Classroom 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.

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

• In Lesson 30, Session 2, Develop, students build conceptual understanding of shapes and understand that they belong in categories by their properties (3.G.A.1).  Students describe the lines and angles in quadrilaterals and triangles and then put them into categories
• In Lesson 12, Fluency and Skills Practice, Using a Multiplication Table, students build procedural fluency as they solve 20 problems, including, “Write 3 possible answers for the equation 36 ÷___=___.”
• In Lesson 19, Session 3, Develop, Try It states, “The Hart School wants to build a new playground. The graph shows the number of dollars each grade has raised to build the playground. Grade 3 and Grade 4 together want to raise $300. How much money must they raise?” The graph shows that Grade 3 has raised $80 and Grade 4 has raised $60.

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

• In the online game platform students are given the opportunity to practice skills that simultaneously work on conceptual understanding and build fluency. For example, the game “Match” focuses on matching multiple representations of multiplication (3.OA.1, 3.OA.7). The cards display products, number sentences, and other visual models to connect multiplication. Students are required to find matches among these representations. 
• In Unit 6, Math in Action, Persevere On Your Own, students engage in application and conceptual understanding to solve,  “At the community center Bella meets an artist who weaves trays. Bella asks the artist to make two snack trays for her. Bella’s ideas are shown below (Two rectangles are displayed). Each tray is shaped like a rectangle. Both trays have the same area. The perimeter of each tray is different. The area of each tray is less than 100 square inches. What size trays can Bella ask the artist to make?” 
• In Lesson 15, Session 2, Develop, students build conceptual understanding of area while using procedural skill of multiplication and division to solve area problems and practice finding areas and side lengths. The problem states, “What is the area of the rectangle?” A picture of a rectangle with a length of 4 cm and a width of 2 cm is shown. Picture It states, “You can use square tiles to find area.” Model It states, “You can also use a multiplication equation to find area.”

The instructional materials reviewed for Ready Classroom Mathematics Grade 3 meet expectations by carefully attending to the full meaning of each practice standard. Overall, the materials attend to aspects of the mathematical practices (MPs) during different lessons throughout the grade, so when taken as a whole, the instructional materials attend to the full meaning of each MP.

In i-Ready, Teach and Assess, Ready Classroom Mathematics, Program Implementation, Standards for Mathematics in Every Lesson, includes information on how the Standards for Mathematical Practice (MPs) are addressed within each lesson, noting that key components of lessons are designed to engage students with the MPs. “Deepen Understanding” provides guidance for teachers on each MP within lessons sessions. Discourse Questions are included, as well as prompts for Structure and Reasoning specifically related to MP7 (Look for and make use of structure) and MP8 (Look for and express regularity in repeated reasoning). In addition, the Try-Discuss-Connect Instructional Routines all identify the MPs. 

The instructional materials attend to the full intent of all eight Mathematical Practices. For example:

MP 1: Make sense of problems and persevere in solving them. 

• In Lesson 13, Session 1, Explore, Try It states, “Kenny has 24 marbles. He puts the same number of marbles into each of 3 bags. How many marbles are in each bag?” Teacher guidance includes “Make Sense of the Problem: To support students in making sense of the problem, have them describe what Kenny decides to do with his 24 marbles.” During Discuss It, students are encouraged to persevere in problem solving. The materials state, “Ask your partner: How did you get started? Tell your partner: I knew… so I …”
• In Lesson 28, Session 3, Develop, Try It states, “Maria has a cooler full of 8 liters of lemonade. She wants to put the lemonade into pitchers to place on the tables at her party. Each pitcher holds 2 liters. How many pitchers does Maria need?” Teacher guidance includes “Make Sense of the Problem: To support students in making sense of the problem, have them identify that the problem is asking them to find the number of 2-liter pitchers Maria can fill with 8 liters of lemonade. Ask How many liters does Maria have in the cooler? How many liters does each pitcher hold?”

MP 2: Reason abstractly and quantitatively

• In Unit 1, Math in Action, Session 1, Deepen Understanding states, “As you discuss the sample solution, point out how the numbers in the solution relate to the problem situation. Encourage students to explain which numbers represent people, which represent dollars, and which represent kits.”
• In Unit 3, End of Unit, Unit Review, Performance Task, students reason quantitatively and abstractly when finding the area of a porch to solve, “Dan is planning to build a square porch attached to the side of his house. After the porch is built, he would like to cover the floor with 1-foot square tiles. The diagram below shows the measurements of the porch and the lawn where he plans to build. How many tiles will he need to cover the porch floor? After Dan bought all of the tiles he needed, he changed his mind about the shape of the porch. How could he change the shape of the porch, but still use the same number of tiles?” Students reason quantitatively when determining factor pairs for a specified area, then reason abstractly when determining which factor pairs are most suitable for the deck. 

MP 4: Model with mathematics.

• In Lesson 17, Session 1, Additional Practice, Problem 3 states, “Write a word problem about this array that you could solve with multiplication or division. Then write an equation to represent your problem.” Students are given a 4 x 6 array of flowers. 
• In Lesson 18, Session 2, Develop, Apply It, Problem 7 states, “Demarco has 4 five-dollar bills. Then his grandfather gives him 1 ten-dollar bill. How much money does Demarco have now? Show your work.” Teacher guidance includes, “For all problems, encourage students to use some kind of model to support their thinking…” There are student work samples that use pictures and models for the problem. 

MP 5: Choose tools strategically.

• In Lesson 15, Session 2, Develop, Try It, students use the available Math Toolkit and can choose from square tiles, grid paper, dot paper, perimeter and area tool (online icon), or multiplication models to find, “What is the area of the rectangle?” In the right margin there is a Math Toolkit, where students are given options to use.
• In Unit 5, Unit Review, Reflect states, “Choose a partner and read the clues for one of your shapes out loud. Have your partner draw the shape they think the clues describe. Does your partner’s drawing match the shape you chose? Explain how the shape you chose and the shape your partner drew can be different, even if your partner did not make a mistake. What tools could you use to make accurate drawings of your shapes? Why would you need each of these tools?”

MP 6: Attend to precision.

• In Unit 6, Math in Action, Session 2, Reflect It, students use precision when discussing, “Be Precise. How did you check that your solution was correct?” with their partner. 
• In Lesson 26, Session 1, students use appropriate symbols when comparing fractions. The Prepare for Using Symbols to Compare Fractions page has students define and give examples for >, <, and =.

MP 7: Look for and make use of structure. 

• In Lesson 7, Session 1, Explore, Try It states, “Katie and Scott are both finding 6 x 7. They each break apart the problem in a different way. Show two different ways to break apart 6 x 7 and find the product.”
• In Lesson 32, Session 3, Develop, Model It states, “Emma drew the rectangle shown (4 x 4 rectangle composed of unit squares). What other rectangles have the same area, but different perimeters?” In Model It, students review a table identifying different perimeters for rectangles with factor pairs of 16. The materials state, “Deepen Understanding, Find Rectangles with a Given Area, SMP7 Look for structure. When discussing the table, prompt students to look for patterns. Ask: ‘Find a pair of rectangles that have the same perimeter. How are their lengths and widths related?’ and ‘Listen for “The 16 x 1 and 1 x 16 rectangles and the 2 x 8 and 8 x 2 rectangles have the same perimeters. In each pair, the length and width are switched.” 

MP 8: Look for and express regularity in repeated reasoning.

• In Lesson 2, Session 3, Develop, Model It, students solve “225 + 229” using regrouping. The materials state, “Deepen Understanding, Regrouping, SMP8 Use repeated reasoning. When discussing the algorithm, prompt students to consider that the steps taken to regroup tens would be the same as those taken to regroup ones. Ask: ‘How many tens are there altogether? How do we regroup the tens? (as 1 hundred and 5 tens) How do we record the tens that have been regrouped as a hundred? How is regrouping tens like regrouping ones? How is it different?’ Listen for “Students should understand that the steps taken to regroup are essentially the same, but regrouped tens are recorded as an extra 1 in the hundreds place and regrouped ones are recorded as an extra 1 in the tens place.”

The instructional materials reviewed for Ready Classroom Mathematics 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. 

In i-Ready, Teach and Assess, Ready Classroom Mathematics, Classroom Resources, the Student Worktext and the Math Journal provide students with opportunities to construct arguments and critique the reasoning of others. Evidence where students have opportunities to construct viable arguments includes:

• In Lesson 2,  Session 1, Explore, Connect It, students solve, “374 + 122.” Problem 3 states, “What is another way you could find 374+122?”
• In Lesson 31, Session 4, Refine, Apply It, Problem 9, Math Journal states, “Jess says that a square cannot be a rectangle because a rectangle has 2 long sides and 2 short sides. Is he correct? Explain.”

Evidence where students have opportunities to analyze the mathematical arguments of others includes:

• In Lesson 13, Session 3, Refine, Apply It, Problem 3 states, “Booth says an odd factor times an odd factor will always equal an even product. Is he correct? Explain.”
• In Lesson 14, Session 3, Develop, Item 2 states, “Anna says the area of this rectangle is 12 square units because each of the small rectangles is 1 unit long. Why is Anna wrong?” 

Throughout the series there are dialogue boxes with the phrase, “Discuss It.” This dialogue box often encourages students to engage in discourse about the mathematics of the lesson. In Lesson 23, Session 4, Develop states, “Ask your partner: Do you agree with me? Why or why not? Tell your partner: I agree with you about...because…..” These prompts require students to analyze their partners’ thinking to determine if they agree or not, and to construct an argument to explain why.

The instructional materials reviewed for Ready Classroom Mathematics Grade 3 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

In i-Ready, Teach and Assess, Ready Classroom Mathematics, Program Implementation, Resources include Discourse Cards which present questions and sentence starters to engage students in mathematical discourse, including the construction of arguments and analysis of others reasoning. For example, “Can you convince your partner or others that your answer makes sense? What do you think about what another student said? Does your partner’s strategy make sense? How is your solution method the same as or different from another student’s method?”

In Classroom Resources, Lesson 0, Understanding the Try-Discuss-Connect Instructional Routine. The Discuss routine presents opportunities to use questions and sentence starters for students to share their thinking and critique each other’s reasoning. In addition, using Compare Strategies, students discuss how representations are the same, different, and related. For example, Lesson 0, Session 2, Compare and Connect, “Ryan has a collection of 284 shells. What is another way to write 284 using numbers? What is another way to write 284 using words?” Teachers have students discuss with partners “How are they the same? How are they different? How are they connected?” These instructional routines are present in every lesson.

Evidence where the instructional materials support teachers to engage students in constructing viable arguments includes:

• In Unit 4, End of Unit, Unit Game, Equivalent Fraction Match states, “Pairs take turns turning over Game cards to find equivalent fractions. Students play until all cards are matched. The student with the most matches wins. Discuss strategies for identifying equivalent fractions.” 
• In Lesson 1, Session 2, Develop, Try It, Discuss It states, “Ask your partner: Do you agree with me? Why or why not? Tell your partner: The strategy I used to find the answer was…” In the teacher’s edition there is a section to support this task titled, “Discuss It, Support Partner Discussion,” which states, “Encourage students to think in terms of the place value they are rounding to as they discuss their solutions.”

Evidence where the instructional materials support teachers to engage students in analyzing the arguments of others includes:

• In Lesson 8,  Session 1, Explore, Try It,  Support Whole Class Discussion states, “Ava’s mom buys two packs of 3 T-shirts.  Her dad buys 3 packs of 2 T-shirts. How many T-shirts did each of Ava’s parents buy? Ask: How do (student name)’s and (student name)’s models show that 2 and 3 are being multiplied two different ways?”

The instructional materials reviewed for Ready Classroom Mathematics Grade 3 meet expectations that materials explicitly attend to the specialized language of mathematics.

In i-Ready, Teach and Assess, Ready Classroom Mathematics, includes several resources to support teachers and students to use the specialized language of mathematics. In Program Implementation, the Academic Vocabulary Glossary identifies the vocabulary, provides a definition, and uses the word in a sample sentence organized by unit. For example, for the term arrange the sample sentence states, “You can arrange the numbers on the place value chart by putting them in their proper columns.” 

In Classroom Resources, Lesson Overview, Lesson Vocabulary identifies whether there is new vocabulary or review, and key terms used in the lesson. For example, Lesson 8, Lesson Overview, “There is no new vocabulary. Review the following key terms. Array is a set of objects arranged in equal rows and columns. Factor is a number that is multiplied.” 

Throughout lessons in Ready Classroom Mathematics, students are provided with Graphic organizers that assist with mathematical language to ensure students are using precise vocabulary. There are five different graphic organizers used throughout the lessons that allows students to organize learning concepts and vocabulary through definitions, illustrations, examples, etc. For example, In the Student Worktext, Lesson 8, Prepare for Using Order and Grouping to Multiply, Support Vocabulary Development, students complete a graphic organizer for factor with “My Definition, My Illustrations, Examples, Non-Examples.”

Build Your Vocabulary is provided at the beginning of each unit to support students in learning and using precise language and terminology. For example, in Unit 3, Beginning of Unit states, “Display, point to, and read each review word aloud. Have students repeat chorally.” Then it suggests for teachers to play “I’m Thinking of a Word” with the Review words. It states, “Read each clue aloud. When you get to the blank snap or clap as a signal to students to write the word you are thinking of in the table.” The teacher reads, “I’m thinking of a word. It is what you do when you want to know how long something is or how tall something is. The word I’m thinking of is ____. What is the word? Write it in the table. (measure).” Once all the clues are done, students discuss their answers and complete the table which has them describe the word. The materials state, “When students are finished, have them read their descriptions aloud. Encourage feedback. Clarify if descriptions are incorrect or incomplete. Have students revise their descriptions.”

Language Objectives are included in the Lesson Overview for each lesson. For example, Lesson 27, Language Objectives include “Use the terms AM and PM appropriately in writing and speaking.”

In the End of Unit, Vocabulary, Vocabulary Cards are available. The materials state, “The purpose of the vocabulary cards is to reinforce students’ understanding of the new vocabulary words in the unit as well as to provide a place for students to record any other math words and definitions that would be helpful for them in their understanding of unit concepts. Students may find it useful to draw or write examples on the vocabulary cards as they encounter the terms during the unit. you may want to make a copy of the cards and display them in a word wall in the classroom to further support students’ learning.”