2nd Grade - Gateway 2

he instructional materials reviewed for Achievement First Mathematics Grade 2 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade-level. Examples include:

• Unit 2, Lesson 7, Introduction and Workshop, students engage with 2.NBT.5, fluently add and subtract within 10 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, as they solve problems about tens and ones by using a variety of representations (stick and dots, expanded form). Students roll two number cubes, record the two-digit number, represent the number using sticks and dots, and represent the number using expanded form. The teacher asks, “How will you figure out how to represent 2-digit numbers using sticks and dots and expanded form?” The students may reply with, “I will look at the digits in each place and think about the value of each digit.” 

• Unit 6, Lesson 2, Introduction, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent the amounts of hundreds, tens, and ones, as they model three-digit numbers with place value blocks, then read and write the numbers. “Kaleb has 3 boxes of 100 crayons, 6 boxes of 10 crayons, and 2 single crayons. How many crayons does Kaleb have?”

• Unit 6, Lesson 14, Introduction, students engage with 2.NBT.4, compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the result of the comparison, as they compare two three-digit numbers written in different forms using <, >, and =. The teacher poses the following comparison problem to students, “562 __ 5 hundreds, 2 tens, 6 ones.” A sample student response, “We wrote 562 > 5 hundreds, 2 tens, 6 ones. We figured it out by showing both numbers in flats, sticks, and dots. For 562 we drew 5 flats, 6 tens, 2 dots. Then for 5 hundreds, 2 tens, 6 ones, we drew 5 flats, 2 sticks, 6 dots. We looked at the hundreds place and saw that they had an equal number of hundreds, so then we looked at the tens and saw that 562 has more tens than 5 hundreds, 2 tens, 6 ones, so 562 is greater than 5 hundreds, 2 tens, 6 ones.” 

• Unit 7, Lesson 2, Aim, students engage with 2.NBT.7, add and subtract within 1000 using concrete models or drawings, as they complete 2-digit addition problems using flats, sticks, and dots. “SWBAT add 2-digit numbers with regrouping in one place by using flats, sticks, and dots.” Workshop Worksheet example, Problem 2A, “$$550 + 268 =$$ ______.”

• Unit 7, Lesson 7, Workshop Worksheet, students engage with 2.NBT.9, explain why addition and subtraction strategies work using place value and properties of operations, as they solve a three-digit subtraction problem, and write a written explanation of why their strategy worked. “Solve. ______ - 246 = 568. Explain how you solved the problem above. What strategy did you use? What steps did you take? Why did your strategy work?”

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include: 

• Unit 2, Lesson 10, Exit Ticket, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, as they add two-digit numbers by using a strategy that makes sense to them (sticks and dots, expanded notation/use known facts). Problem 1, “$$62 + 27 =$$ ___.” Students are directed to use sticks and dots or expanded notation to solve. 

• Unit 3, Practice Workbook B, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, as they identify the proper model for a given problem. Problem 34, “Circle which set of sticks and dots will help to find the total? 62 + 24 = ______.” 

• Unit 6, Lesson 2, Independent Practice Worksheet, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent the amounts of hundreds, tens, and ones, as they independently model three-digit numbers with place value blocks, then read and write the numbers. Problem 1, “Draw flats, sticks, and dots to represent each number. 258. How many more ones will make a ten? How many more tens will make a hundred? How many more hundreds will make a thousand?”

• Unit 7 Lesson 7, Exit Ticket, students engage with 2.NBT.9, explain why addition and subtraction strategies work using place value and properties of operations, as they solve a three-digit subtraction problem, and write a written explanation of how they solved the problem. Problem 2, “_____ - 567 = 293. Explain how you solved.”

• Unit 7, Lesson 10, Independent Practice, students engage with 2.NBT.7, add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction and relate the strategy to a written method. Problem 2, “Solve using flats, sticks, and dots. 531 - 258 = ____.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. These skills are delivered throughout the materials in the use of games, workshop, practice workbook pages and independent practice, such as exit tickets. 

The materials develop procedural skill and fluency throughout the grade-level. Examples include: 

• Unit 2, Lesson 2, Introduction and Workshop, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they determine the missing part to make 10 using a strategy that works for them (count up, count back, just know). Students use a dot cube to roll for a number to subtract from 10 in a number bond. Potential strategy examples, “Count up: You can start at 6 because that’s the first part and count up until 10 because that’s the whole. Like this… 6 -- 7, 8, 9, 10. So the missing part is 4. Subtract: You can start with the whole -- 10 and subtract 6 because that’s the part we know. The answer is 4, so the missing part is 4. Count back: I started at 10 because that is the whole and then I counted back 6 because that’s the part we know. Like this 10 -- 9, 8, 7, 6, 5, 4. So the missing part is 4. Just know: I just know that 6 and 4 make 10 because they’re number pairs. So the missing part must be 4.” 

• Unit 3, Practice Workbook B, Activity: Building Toward Fluency, students engage with 2.OA.2, fluently adding and subtracting within 20 using mental strategies, as they use various strategies to complete and discuss addition problems. “Write the expression on the board or chart paper. Start with 4 + 10. Ask students to describe their strategy for solving the problem. Choose one or more students to explain their strategy to the class. Represent each strategy on the board using the number line or magnetic cubes. Once the student’s strategy is understood by the class, continue with the next sum.”

• Unit 5, Practice Workbook B, Ten Plus Number Sentences, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or relationship between addition and subtraction, as they practice proficiency with their ten plus facts. The teacher says, “I will flash two ten-frame cards, ten and another card. Wait for the signal. Then tell me the addition sentence that combines the numbers.” The teacher flashes a 10 and 5. Students respond with, “$$10 + 5 = 15$$”

• Unit 6, Lesson 9, Workshop Worksheet, students engage with 2.NBT.2, skip-count by 5s, 10s, and 100s, as they use skip counting by 10s and 100s to count up. Problem 1, “Count from 90 to 300.”

The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include: 

• Unit 2, Lesson 3, Exit Ticket, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they independently complete a number bond with one unknown number and write 2 addition and two subtraction problems to match. Problem 1, “Finish the number bond and write number sentences to match.” Students are provided with a number bond diagram with 11 and 6 as addends and an unknown sum.

• Unit 3, Practice Workbook B Pairs To Make Ten With Number Sentences, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they solve doubles +2 facts. Problem 3, “$$2 + 4 =$$ ___.”

• Unit 4, Practice Workbook B, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations and/or the relationship between addition and subtraction, as they independently solve two-digit addition and subtraction problems. Problem 21, “$$53 -$$ ______ = 28.”

Unit 8, Practice Workbook D, students engage with 2.NBT.8, mentally add 10 or 100 to a given number 100-900, as they independently add 10 or 100 to given numbers betwembers under 1000. Problem 1a, “Solve each problem using mental math, 678 + 100 = ____.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Students are given multiple opportunities to engage in real-world applications, especially during Math Stories, which include both guided questioning and independent work time, and Exit Tickets to independently show their understanding.

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 1, Guide to Implementing AF Math, Math Stories, September, students engage with 2.G.1, recognize and draw shapes with specified attributes, in a non-routine problem. Sample Problem 3, “Corie is making three-dimensional shapes in art class. He has made 18 faces in total. What kind of shapes could he have made and how many of each?”

• Unit 3, Guide to Implementing AF Math, Math Stories, December, students engage in a routine problem with 2.OA.1, using addition and subtraction within 100 to solve one- and two-step word problems, as they calculate take from problems with results unknown. Sample Problem 2, “Antonio gave 27 tomatoes to his neighbor and 15 to his brother. He had 72 tomatoes before giving some away. How many tomatoes does Antonio have remaining?”

• Unit 8, Guide to Implementing AF Math, Math Stories, May, students engage with 2.OA.4, using addition to find the total number of objects in a regular array, in a non-routine problem. Sample Problem 15, “A tic-tac-toe board has 3 columns with 3 rows in each. How many different ways could someone win the game (get 3 in a row)? ”

Materials provide opportunities for students to independently demonstrate routine and non-routine applications of the mathematics throughout the grade level. Examples include: 

• Unit 3, Lesson 7, Independent Practice, students engage in a routine word problem with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they solve a story problem. Problem 7, “Enmaries has a jump rope that is 68 inches long. Giada’s is 33 inches shorter than Enmaries’s jump rope. What is the length of Giada’s jump rope?”

• Unit 5, Lesson 3, Independent Practice, students engage in a routine word problem with 2.MD.8, solve word problems involving money, as they independently solve word problems with money. Problem 5, “King Jamonie has 3 quarters, 1 dime, 2 nickels, and 4 pennies. How much money does he have?” 

• Unit 6, Lesson 17, Workshop, students engage in a non-routine word problem with 2.NBT.4, compare two three-digit numbers. “Nehemiah and Sonya are exercising to stay healthy. In the month of January, Nehemiah exercised 198 minutes for the first week and 277 the second week. Sonya exercised 309 the first week and 172 the second week. Write a comparison statement to show who exercised more. Show all of your mathematical thinking.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

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

Conceptual understanding

• Unit 6, Lesson 2, Exit Ticket, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent the amounts of hundreds, tens, and ones, as they read and write numbers within 1,000 after modeling with place value blocks (flats, sticks, and dots). Problem 2, “Draw models of ones, tens, and hundreds.” Students are given the number 508 and asked to answer the following questions, “How many more ones will make a ten? How many more tens will make a hundred? How many more hundreds will make a thousand?”

• Unit 6, Lesson 3, Independent Practice, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones, and 2.NBT.3, read and write numbers to 1000, as they represent a three-digit numbers in a variety or forms and models. Problem 10, “Alexander has 529 M&Ms. Write the amount of M&Ms Alexander has in three different ways by filling in the blanks. (Unit Form, Base Ten Numeral Form, Place Value Models)”

• Unit 7, Lesson 1, Independent Practice, students engage with 2.NBT.7, add and subtract within 1000 using concrete models or drawings and strategies based on place value, as they independently solve a three-digit subtraction. The third item, “Solve using flats, sticks, and dots ______ - 348 = 650. Explain how you solved ______ - 348 = 650.”

• Unit 9, Practice Workbook E, students engage with 2.NBT.7, add and subtract within 1000 using concrete models or drawings and strategies based on place value, as they independently solve a three-digit addition problem using a number line. Problem 4, “Use the number line to solve. Show your work. 578 + 237 = ___.” Students are provided a blank number line. 

Procedural skills (K-8) and fluency (K-6)

• Unit 2, Lesson 4, Exit Ticket, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they solve addition problems. Problem 1, “Solve. 7 + 8 = _____.”

• Unit 2, Lesson 24, Exit Ticket, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition/subtraction, as they solve two-digit subtraction problems with missing minuends by relating addition and subtraction. Problem 1, “Solve. 93 - ____ = 62.” Students are provided with a blank number bond model.

• Unit 3, Practice Workbook B, students engage with 2.NBT.5, fluently adding and subtracting within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, as they solve put together problems with an unknown addend. Problem 5, “$$35 +$$ ___ = 50.”

• Unit 5, Practice Workbook B, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or relationship between addition and subtraction, as they practice fluently adding and subtracting. Problem 4, “$$45 +$$ ___ = 100”

Application

• Unit 2, Guide to Implementing AF Math, Math Stories, October, students engage with 2.OA.1, adding and subtracting within 100 to solve one- and two-step word problems, as they solve addition problems with the change unknown. Sample Problem 2, “Jose has 27 erasers. Kate gave him some more. Now he has 53 erasers. How many erasers did Kate give him?”

• Unit 3, Lesson 4, Independent Practice, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they solve a story problem. Problem 4, “Ms. Reinhardt has 42 books. Ms. Gomez has 18 fewer books than Ms. Reinhardt. How many books does Ms. Gomez have?”

• Unit 3, Lesson 4, Exit Ticket, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems, as they independently solve compare/smaller unknown word problems. Problem 1, “There are 59 girls on the bus. There are 26 more girls than boys on the bus. How many boys are on the bus?”

• Unit 5, Lesson 8, Exit Ticket, students engage with 2.MD.8, solve word problems involving dollar bills, quarters, dimes, nickels and pennies, using and $$\cancel{C}$$, as they solve one-step story problems of all types that involve bills and coins by using the most efficient strategy. Problem 2, “Kevin has 75 cents. He spent 3 dimes, 3 nickels, and 4 pennies on a slice of cake. How much money does he have left?”

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

• Unit 2, Lesson 6, Exit Ticket, students engage with 2.NBT.9, explaining why addition and subtraction strategies work, using place value and the properties of operations, and 2.OA.1, adding and subtracting within 100 to solve one- and two-step word problems, as they solve take from problems with the result unknown (application) and show their thinking (conceptual understanding). Problem 1, “Represent and solve. Amya has 17 pencils. 13 are red and the rest are green. How many green pencils does Amya have? Describe how you solved.”

• Unit 3, Lesson 2, Independent Practice, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, and 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they determine if a representation is correct (application) and how they know (conceptual understanding). Problem 10, “Mr. Johnson has 46 pens. 24 are blue and the rest are black. How many of Mr. Johnson’s pens are black? Charlie and Henry represented the problem below. (Charlie 46 + 24 = ? represented/ Henry 46 - 24 = ? represented)”

• Unit 5, Lesson 8, Exit Ticket, students engage with 2.MD.8, solve word problems involving money, as they represent (conceptual understanding) and solve story problems (application) using the most efficient strategy (procedural skill). Problem 1, “Jacob bought a piece of gum for 26 cents and a newspaper for 61 cents. He gave the cashier $1. How much money did he get back?”

• Unit 8, Lesson 2, Independent Practice, students engage with 2.OA.4, use addition to find the total number of object arranged in a rectangular array, as they draw a rectangular array and write addition equations (conceptual understanding) to represent and solve word problems (application) involving equal groups of objects. Problem 7, “Ja-yier put 5 toys into 4 different baskets. How many toys does Ja-yier have in all?”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson. 

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Examples include:

• Unit 2, Lesson 6, Narrative, What is new/or hard about the lesson? provides an explanation to the teacher about how the content engages with MP1. “This lesson is challenging because it’s pushing students to apply their understanding of part-part-whole relationships and pushing them to become fluent within 20. Students may have difficulty understanding the context of the story problem. They may also have difficulty calculating fluently.” Introduction, Pose the Problem, “Carla baked 15 desserts. 9 of them were chocolate chip cookies and the rest were brownies. How many brownies did Carla bake?” 

• Unit 4, Assessment, students make sense of data presented in a graph to solve two-step story problems. Problem 4, “19 of the scholars who like fruit are girls. How many of the scholars are boys?” Students are provided a bar graph showing survey data regarding favorite fruits.

• Unit 8, Lesson 8, Introduction, students make sense of even/odd. “There are 6 students in Mr. Johnson’s reading group. If he wants his students to work in pairs, will everyone have a partner? Use pictures to prove your thinking.” 

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Examples include:

• Unit 1, Lesson 7, Narrative, “Students also engage with MP 2 by reasoning abstractly and quantitatively when they discuss and explain how much longer/shorter one line is than the other line. Students’ understanding of quantities will help then determine if their calculations are reasonable by determining if their answer makes sense. Students use reasoning of abstract length through questions like ‘how can we find out how long is Line A that Line B?’ and ‘How does this work?’”

• Unit 5, Lesson 5, Independent Practice Worksheet, students reason through story problems involving coins, requiring them to abstract the value of a set of coins, to find the total value. Problem 4, “Enrique has 3 quarters, 1 dime, 2 nickels, and 4 pennies. How much money does he have?”

• Unit 7, Lesson 15, Narrative, “Students engage with MP 2 as students reason abstractly when they discuss the relationship between subtraction and addition and the part-part-whole relationship to solve problems with unknowns in all positions.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Examples include:

• Unit 2, Lesson 3, Introduction, students are prompted to analyze the thinking of others as they determine the mistakes made by the teacher as he/she models how to find the unknown number in a number bond. “(Model an intentional mistake--add when you should subtract.) (Show number bond with 9 in the center/whole and 5 in the part--label 9 red and blue crayons, 5 red crayons, missing # blue crayons.) Oh! I can find the missing number by adding 5 and 9. What do you think? What mistake did I make? How can I fix it?”

• Unit 3, Lesson 2, Introduction, Problem 2, “Khaleel and Mauricia represented the problem below. There were 36 kids on the playground. Some more kids came over to join them. Now there are 62 kids on the playground. How many kids came to join them? Look at Khaleel and Mauricia’s representations. Who is correct? How do you know?” 

• Unit 5, Lesson 5, Independent Practice, Check for Understanding, “Why did you trade ___ for ___? What strategy did you use to make trades?”

• Unit 7, Lesson 19, Assessment, Problem 4, “Find the missing numbers to make each statement true. Show your strategy to solve. a. ___ = 407 - 159. Explain how you solved this using what you know about place value.”

• Unit 8 Assessment, students use their knowledge of arrays to analyze the mathematical thinking of a fictitious student. Item 1, “Angela wants to make 3 pins. Angela wants to put 5 beads on each pin. Angela has a bead box with three rows in it. Each row has five sections. Angela has one bead in each section. Angela says that she has enough beads to make three pins. Is Angela correct? Show all of your mathematical thinking.”

• Unit 9, Lesson 4, Intro, Problem 2, “Chase offers to share his pie with Jariah and Luke. They want to have the largest pieces possible. Should Chase cut it into thirds or fourths? Why?”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

There is intentional development of MP4 to meet its full intent in connection to grade-level content. Examples include:

• Math Stories Guide, Promoting Reasoning through the Standards for Mathematical Practice, MP4, “Math Stories help elementary students develop the tools that will be essential to modeling with mathematics. In early elementary, students become familiar with how representations like equations, manipulatives, and drawings can represent real-life situations.” Within the K-4 Math Stories Representations and Solutions Agenda, students are given time to represent, retell, and solve the problem on their own.

• Unit 3, Lesson 2, Narrative, “In this lesson, scholars will focus on accurately representing the problem and then using that representation to choose the correct operation to solve. They will work with Add To/Take From - Change Unknown and Put Together/Take Apart - Addend Unknown story problems.” Introduction task, “There are some birds on a fence. 19 birds flew away. Now there are 52 birds on the fence. How many birds on the fence were there to start?"

• Unit 8, Lesson 6, Narrative, “Students engage with MP4 as they model with math tiles and drawings to analyze the relationship between rows and columns of arrays to rectangular arrays. The use of the math models to help deepen their understanding of rectangular arrays and why they work.” Pose the Problem, “Use your tiles to make a rectangular array with 12 total squares and 4 columns. Then draw the array and write a repeated addition sentence to match.”

• Unit 9, Lesson 8, Exit Ticket, “Leani and Carla each baked a cake in the same rectangular pan. Leani ate one-fourth of her cake. Carla ate one-third of her cake. a. Show how Leani and Carla cut their cake. Share the fraction that they ate. Who ate more cake? How do you know?” Narrative, “Students also engage in MP4 through the story problem protocol to use mathematical models and connect it back to the story problem. Students model the story problem with an appropriate representation and use an appropriate strategy to explain their answer. Students also engage with MP4 as they determine if their answer makes sense and connect their answer back to the story problem by finishing the story. This process occurs with each story problem they solve throughout the lesson.”

There is intentional development of MP5 to meet its full intent in connection to grade-level content. Examples include:

• Unit 1, Lesson 6, Narrative, “Students engage with MP 5 as they explain and justify which math tools to use for estimating the measurements either with centimeters or a meters. Students will decide and justify which mental benchmarks of an M&M is about the same size as a centimeter and their opposite shoulder to fingertips is about the same length as a meter to estimate the best measurement as they solve each problem throughout the lesson.” 

• Unit 2, Lesson 23, Narrative, “Students engage with MP5 as they solve problems with missing addends using a strategy of their choice: sticks and dots, expanded notation, or a number line to solve. The teacher and students make connections between the various strategies and help students become more efficient in solving throughout the lesson.”

• Unit 6, Lesson 12, the Exit Ticket shows a table with the columns from left to right having 100 less, 10 less, Starting Number-565, 10 more, 100 more. Students are to complete the table and are not guided to use any specific tools or strategies. “Students engage with MP5 by using a strategy to represent the problem that works best for them to add or subtract by 10 or 100.” In the lesson, the first problem of the bingo game, “How can I represent 38 + ____ = 72 with a number bond? How can we figure out the missing part? We can use a number line and count up, we could subtract using sticks and dots or expanded notation.”

At times, the materials are inconsistent. The Unit and Lesson Overview narratives describe explicit connections between the MPs and content, but lessons do not always align to the stated purpose. 

The materials do not provide students with opportunities or guidance to identify and use relevant external mathematical tools and resources, such as digital content located on a website.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

There is intentional development of MP6 to meet its full intent in connection to grade-level content. Many problems present students with the opportunity to attend to precision within the mathematics and the reasoning of the answer. Examples include: 

• Unit 1, Lesson 1, Introduction, students attend to precision with measurement. “How long is it? How can I measure the crayon? Note: Teacher should model each measuring strategy as it is named. Measure the longest part: When we measure length we measure the LONGEST part of the object. Line up the endpoints.” Students might say, “You need to start at one end and go to the other end. No gaps or overlap. You need to make sure the cubes are right next to each other and there aren’t gaps or overlaps.”

• Unit 4, Lesson 2, Introduction, students attend to precision when categorizing items in a picture graph. “Why is it important to draw all your pictures the same size? It is important to draw all the pictures the same size because it makes it easier to read the graph. If all the pictures are the same size we can easily tell which group has more or less.” 

• Unit 9, Lesson 3, Introduction, students attend to precision when differentiating between parts using the concept of division. “Yesterday you used pattern blocks to divide shapes into equal parts-halves, thirds, and fourths. Teach your partner what you know about halves, thirds, and fourths. Halves means 2, the whole is divided into 2 equal parts, thirds means the whole is divided into 3 equal parts, fourths means the whole is divided into 4 equal parts.”

The instructional materials attend to the specialized language of mathematics. The materials use precise and accurate mathematical terminology. Examples include:

• Unit 4, Lesson 3, Introduction, teachers use accurate terminology to create bar graphs. “You know how to sort objects into categories to create tally charts and pictographs. Today we are going to use those categories to make bar graphs. Before we can make a bar graph what are some things we need in our graph? We need a title, categories, category labels, scale, scale labels, and bars! The scale is the number on the side of the bar graph that tells us how many in each category. How do I know where to stop on my scale? You need to find the category with the largest number, then you need to make sure the scale goes up to that number so that all of your data fits.”

• Unit 8, Lesson 3, Introduction, teachers provide explicit instruction in the definition of an array, horizontally, and vertically, as they organize objects of equal groups into rows and columns. “An array has rows that go horizontally, or side to side, and it has columns that go vertically, or up and down. In an array, all of the rows are equal and all of the columns are equal. We can think of the rows as our groups and the columns help us see how many we have in each group.”

• Unit 9, Overview, Major Misconceptions & Clarifications, “Misconception: Students confuse numerator and denominator. Clarification: Have students label their fraction with words. The numerator as the part and the denominator as all of the parts.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 2, Lesson 13, Narrative, “Students will learn the structure within the number line representation to explain why and how it works.”

• Unit 6, Lesson 12, Narrative, “Students also engage with MP 7 as they discuss and make connections between the place values and adding/subtracting by 10 or 100. Students are able to recognize that adding/subtracting by 10 changes the number on the tens place by 1 and adding/subtracting by 100 changes the digit in the hundreds place by 1.” Exit Ticket Problem 1 has a table with the following listed from left to right: “100 less, 10 less, starting number -565, 10 more, and 100 more.”

• Unit 8, Lesson 7, Workshop Worksheet, students decompose numbers and apply repeated addition to the structure of arrays to both the rows and columns. Question 1, “Jeremiah drew an array with 20 squares. Draw 3 different arrays that have 20 squares in all. Write a repeated addition sentence to match each array you drew.”

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 7, Lesson 4, Narrative, “Students continue to deepen their understanding of the structure of adding starting with the correct place and also recognizing the repeated reasoning and calculations involved in regrouping when there are 10 or more in any place to make a bundle of a ten or a hundred when adding.”

• Unit 8, Lesson 10, Introduction, Pose the Problem, students engage in repeated reasoning. “(Have VA (visual aid) with #s 1-20 written across the top, drawing of 2 empty 10-frames.) How many counters do we have? Is 2 an even number? How do you know? How many counters now? Is 3 even or odd? How do you know? How many counters altogether? Is 4 odd or even? How do you know? What numbers on the poster are even? How do you know? What do you notice about the numbers we circled? Do you see a pattern?”

• Unit 9, Lesson 7, Narrative, “How does the learning connect to previous lessons? What do students have to get better at today? In the previous lesson, students partitioned shapes into the same fraction in more than one way and came to the understanding that the same fraction can have a different shape. Students also named and wrote unit fractions. Today, for the first time, students will partition rectangles into fractions in more than one way and prove the fractions are the same by cutting the parts and manipulating the pieces.”