1st Grade - Gateway 2

The instructional materials for Achievement First Mathematics Grade 1 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 3, Lesson 5, Introduction, students engage with 1.OA.3, apply properties of operations as strategies to add and subtract, 1.OA.4, understand subtraction as an unknown-addend problem, and 1.OA.5, relate counting to addition and subtraction, as they represent addition and subtraction scenarios with number bonds. The teacher models how to play Roll and Record: Mixed Operations. “Step 1: Pick a card and roll 2 cubes. (pick addition operation card first - for planning purposes, assume you roll 4 and 6), Step 2: solve and record equation. What’s the total?, Step 3 says record with a number bond; label parts and whole, Step 4 says record with other operation equation. We already wrote an addition equation… so now we need to record with subtraction.” 

• Unit 5, Lesson 6, Introduction, students engage with K.CC.6, identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, as they play a game called Compare. During the Introduction, two cards are drawn (example, 7 and 9) and students are asked to pictorially show which is more or less by drawing circles on their whiteboards. The teacher asks, “How do you know from the picture?” A sample student response might be, “I know because in the picture you can see that there are extra circles in the row of 9 and the row of 7 is missing some.” 

• Unit 6, Lesson 2, Workshop, students engage with 1.NBT.2, understand that the two digits of a two-digit number represent amounts of tens and ones, as they select a bag with 10-90 cubes and draw a representation of the two-digit numbers, showing tens and ones. The teacher is provided support in the Assessment and Criteria for Success portion of the lesson, “Students will pick a bag that is filled with ten sticks and loose ones. They will determine how many by counting by tens and ones and draw a literal picture and write a numeral to match. Students should be able to explain why they are counting by tens and ones and what their picture and numeral represents. For example, ‘In my picture I drew 7 ten sticks and can count them by ten because there are ten cubes in each stick. Then I draw 4 loose ones and I would count on by ones. So “10, 20, 30, 40, 50, 60, 70, 71, 72, 73, 74.” There are 74. I would write 7 to show 7 groups of ten and 4 to show 4 loose ones.’”

• Unit 6, Lesson 20, Introduction, students engage with 1.NBT.3, by comparing two two-digit numbers based on the meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, <. Step 5, “Is 65 greater or less than 63? How can we figure it out? Strategy 1: Sticks and dots (intervention). SMS: We could look at the sticks and dots and it looks like 65 has the same number of sticks/tens as 63 but 65 has more ones than 63. Therefore, 65 is greater than 63.”

• Unit 9, Lesson 12, Workshop Worksheet, students engage with 1.NBT.6, subtract multiples of 10 in the range 10-90, using models, drawings, and strategies based on place value, as they subtract multiples of 10 from a two-digit number using strategies that work for them. Problem 4, “$$50 - 30 =$$ _____. How did you solve?”

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

• Unit 3, Lesson 11, Exit Ticket, students engage with 1.OA.1, use addition and subtraction within 20 to solve word problems, as they solve story problems by visualizing and representing in a way that makes sense to them. Problem 2, “Tony was collecting buttons. He had 4 buttons and then his grandmother gave him 3 more buttons. How many buttons does he have now?”

• Unit 6, Lesson 9, Exit Ticket, students engage with 1.NBT.4, add within 100, including adding a two-digit number and a one-digit number and adding a two-digit number and a multiple of ten, as they combine two multiples of ten by using a strategy that makes sense to them (cubes, literal pictures, sticks and dots, count all/on by tens, use place value). Problem 1, “Solve. 30 + 20 = ____.”

• Unit 7, Practice Workbook D, students engage with 1.NBT.2, by understanding that the two digits of a two-digit number represent amounts of tens and ones. Problem 12, “Show the number 39 in tens and ones.”

• Unit 7, Practice Workbook D, students engage with 1.NBT.6, subtract multiples of 10 in the range 10-90, using models, drawings, and strategies based on place value, as they independently subtract multiples of 10 from a two digit number using strategies that work for them. Problem 1, “$$80 - 60 =$$ _______.”

• Unit 9, Lesson 12, Exit Ticket, students engage with 1.NBT.6, subtract multiples of 10 in the range 10-90 from multiples of 10 in the range of 10-90, as they use strategies that work for them (count what’s left, count back, uses known facts). Problem 1, “Solve. 50 - 30 = __.” Additional guidance for the teacher is found in Assessment and Criteria for Success, “Students should be able to describe their work by saying, ‘I solved 50 - 30. First I drew 5 sticks and 0 dots to show 50 because there are 5 tens and 0 ones. Then I need to take away 30, which is 3 tens and 0 ones. So as I crossed out the sticks I counted back like this. 50 -- 40, 30, 20. The difference is 20.’”

The materials for Achievement First Mathematics Grade 1 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 but are not limited to: 

• Unit 3, Lesson 9, Assessment and Criteria for Success, students engage with 1.OA.3, apply properties of operations as strategies to add and subtract, and 1.OA.4, understand subtraction as an unknown-addend problem, as they explain how they found the parts of their total. Workshop Written Assessment, “Students find the number pairs to make a total by guessing and checking, counting up, counting back, or using known facts. Exemplar Student Response, “My total is 6. I found the parts by picking a card and then counting up to the total. So I picked a 4. Then I counted up until I got to 6 because that’s the whole. I got 2 so that’s the other part. Then I recorded by putting 6 here because it’s the whole. Then I put 4 and 2 here because they are the parts. I wrote the equation 4 + 2 because I’m combining the parts = 6 because they make the whole.”

• Unit 3, Practice Workbook B, Activity: X-Ray Vision, students engage with 1.OA.6, adding and subtracting within 20, demonstrating fluency for addition and subtraction within 10, as they calculate the missing addend using counters. Partners to 10, “Place 10 counters on the floor next to a container. Tell students to close their eyes. Put one of the items into the container. Tell students to open their eyes and identify how many counters were put inside it. Continue the game, eliciting all partners to 10.”

• Unit 4, Practice Workbook B, Activity: Ten and Tuck, students engage with 1.OA.6, add and subtract within 20, as they use their fingers to make 10. “Directions: Tell students to show 10 fingers. Instruct them to tuck three (students put down the pinky, ring finger, and middle finger on their right hands). Ask them how many fingers are up (7) and how many are tucked (3). Then, ask them to say the number sentence aloud, beginning with the larger part (7 + 3 = 10), beginning with the smaller part (3 + 7 = 10), and beginning with the whole (10 = 3 + 7 or 10 = 3 + 7).”

• Unit 5, Lesson 5, Workshop, Intro Packet, students engage with 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10, as they add three numbers rolled with number cubes, using the strategy of grouping facts they know or can easily figure out. Problem 1, “$$5 + 3 + 5$$.”

The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include but are not limited to:

• Unit 3, Lesson 4, Exit Slip, students engage with 1.OA.6, add and subtract within 20, as they use number bonds to help them solve. “Find the difference between the number cubes. Represent by completing the number bond and equation.” Cubes show the numbers 9 and 6. 

• Unit 4, Practice Workbook B, Math Sprint A, students engage with 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10, as they practice addition and subtraction facts on a Math Sprint. Problem 23, “___ - 6 = 3” 

• Unit 5, Lesson 21, Exit Slip, students engage with 1.OA.6, adding and subtracting within 20, demonstrating fluency for addition and subtraction within 10, as they find the missing subtrahend of a subtraction equation. “Fill in the blank to make the equations true. 10 - ___ = 8 - 2.”

• Unit 6, Lesson 17, Exit Ticket, students engage with 1.NBT.5, given a two-digit number, mentally find 10 more without having to count, as they independently add ten to a number. Problem 1, “$$32 + 10 =$$____.”

The materials reviewed for Achievement First Mathematics Grade 1 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 1.OA.2, represent and solve addition and subtraction problems within 20, in a non-routine problem. Sample Problem 3, “Carla is making fruit salad. She uses 8 apples and 2 more bananas than apples. How many pieces of fruit has she used altogether? ”

• Unit 2, Guide to Implementing AF Math, Math Stories, October, students engage with 1.OA.1, use addition and subtraction within 20 to solve routine word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions. Sample Problem 13, “There were 20 kids at the haunted house. Then, some ran out screaming. Now there are only 5 kids in the haunted house. How many kids ran out screaming?”

• Unit 3, Guide to Implementing AF Math, Math Stories, November/December, students engage with 1.G.1, identifying shapes, in a non-routine problem. Sample Problem 3, “Sammy grabs a handful of pattern block shapes. He gets 2 triangles, 1 trapezoid, 2 squares, and 1 hexagon. Ari grabs a handful too. He gets 3 rhombuses, a rectangle and 2 triangles. They each count to see how many 4-sided shapes they got. Who got more four-sided shapes? Ari or Sammy?”

• Unit 5, Guide to Implementing AF Math, Math Stories, February, students engage in a non-routine problem with 1.OA.1, solve addition and subtraction word problems within 20, as students calculate take apart problems with both addends unknown. Sample Problem 3, “Ms. Russo had 20 awards to pass out to her class. Her class has boys and girls. How many could she pass out to the girls? (after they represent: Find at least 4 different solutions) ($$0 + 20$$ is a solution).”

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

• Unit 4, Workbook E, students engage with 1.MD.4, interpreting data with up to three categories and answering questions, and 1.OA.2, solving addition problems of three whole numbers with a sum less than 20, as they calculate routine add-to problems with the results unknown of three addends in a non-routine problem. Problem 11, “The class has 18 students. On Friday, 9 students wore sneakers, 6 students wore sandals, and 3 students wore boots. Use squares with no gaps or overlaps to organize the data. Write a number sentence to tell how many students were asked about their shoes on Friday.”

• Unit 5, Lesson 23, Lesson 15 Task, students engage with 1.OA.7, understand the meaning of the equal sign and determine if equations involving addition are true or false, as they solve a routine word problem asking them to determine if the total of two groups are equal. Problem 1, “Ben has 9 ladybugs and 5 crickets in his jar. Jill has 8 ladybugs and 7 crickets in her jar. Dad thinks they have the same amount of insects in each jar. Is Dad correct? Show and tell how you know.” 

• Unit 8, Lesson 8, Exit Ticket, students engage with 1.MD.2 to solve a routine word problem, by expressing the length of an object as a whole number of length units by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. “How much longer is pencil B than pencil A? Use your inch tiles to help you. Pencil B is ___ inch tiles LONGER than pencil A.”

The materials reviewed for Achievement First Mathematics Grade 1 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 13, Exit Ticket, students engage with 1.NBT.6, subtract multiples of 10 in the range of 10-90 from multiples of 10 in the range 10-90, as they represent and solve two subtraction problems on an exit ticket. Problem 1, “Represent and solve. 50-30” Problem 2, “Represent and solve. 90-40” Assessment and Criteria for Success, “Students will find the difference of two multiples of ten. They may use any strategy that works. If counting back with fingers, they should be able to explain, ‘I started with 90 and then counted back 40 by counting back by tens 4 times because 40 is 4 tens.’”

• Unit 7, Practice Workbook D, students engage with 1.NBT.2, understand that the two digits of a two-digit number represent amounts of tens and ones, as they write the number represented by images of sticks and dots. Problem 5, “Which number is represented?” Four rods are shown with five dots.

• Unit 9, Lesson 9, Introduction, students engage with 1.NBT.4, add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of ten, as they use a strategy that makes sense to them (cubes, sticks and dots, count on by tens, expanded notation addition). Students pick two cards such as 47 and 50 and find the total. Students are to choose a strategy such as “count all by tens and ones.” The students might say, “We can count all of the tens by ten and the ones by one. Like this, 10, 20, 30, 40, 50, 60, 70, 80, 90, 91, 92, 93, 94, 95, 96, 97).” Students may also choose to count on by tens and ones. A student might explain, “I just know that we have 47 right there, so then I can just count on by tens like this 47 -- 57, 67, 77, 87, 97 (can use tens sticks with cubes or sticks to help count on).” 

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

• Unit 3, Lesson 2, Exit Slip, students engage with 1.OA.6, adding and subtracting within 20, as they use number bonds to build addition equations. Problem 1, “Find the total of the number cubes. Represent by completing the number bond and the equation.” Two cubes are shown with 5 and 6 on them. A number bond frame is provided and “___ + ___ = ___.”

• Unit 4, Practice Workbook B, Number Bond Roll, students engage with 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10, as they review number bonds allowing students to build and maintain fluency with addition and subtraction facts within 10. “Match partners of equal ability. Each student rolls one die. Students use the numbers on their own die and their partner’s die as the parts of a number bond. They each write a number bond, addition sentence, and subtraction sentence on their personal white boards.”    

• Unit 5, Lesson 13, Exit Slip, students engage with 1.OA.6, add and subtract within 20, as they solve addition and subtraction problems by creating a fact family. “Find the rest of the fact family. 13 - 6 = 7” 

Application

• Unit 2, Guide to Implementing AF Math, Math Stories, October, students engage with 1.OA.1, adding and subtracting within 20 to solve word problems. Sample Problem 12, “Zamira read 18 books. Some were about bugs. 2 were about snakes. How many books about bugs did she read?”

• Unit 3, Lesson 13, Exit Ticket, students engage with 1.OA.1, use addition and subtraction within 20 to solve 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 2, “I made 3 yellow paper chains and 5 blue paper chains. How many paper chains did I make?”

• Unit 4, Guide to Implementing AF Math, Math Stories, January, students engage with 1.OA.1, adding and subtracting within 20 to solve word problems. Sample Problem 10, “Diego wrapped 24 presents. Jessica wrapped 9. How many fewer presents did Jessica wrap than Diego?”

• Unit 5, Guide to Implementing AF Math, Math Stories, February, students engage with 1.OA.1, adding and subtracting within 20 to solve word problems, as they solve compare problems with the smaller number unknown. Sample Problem 13, “Shayla has 12 fewer pencils than Matthew. Matthew has 19 pencils. How many pencils does Shayla have?”

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 3, Lesson 9, Exit Ticket, students engage with 1.OA.1, using addition and subtraction within 20 to solve word problems, and 1.OA.6, adding and subtracting within 20, as they solve take apart problems (application) with both addends unknown (conceptual understanding). Problem 2, “There were 7 animals on the farm. Some were sheep and some were pigs. How many could be sheep and how many could be pigs? Show one combination using a number bond and an equation.”

• Unit 3, Lesson 19, Exit Ticket, students engage with 1.OA.1, use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions and 1.OA.6, add and subtract within 20, as they solve a story problem (application), as they solve problems within 20 (procedural skill). Problem 5, “Sarah has 9 pennies. Michael has 6 pennies. How many more pennies does Sarah have than Michael? You can use your cubes to help you solve.” Assessment and Criteria for Success, Exemplar Response, “Sara has 3 more pennies than Michael. I know because I built 9 cubes and 6 cubes and put them next to each other. They both have 6 cubes but Sarah has 3 more pennies than Michael.” 

• Unit 4, Lesson 5, Exit Ticket, students engage with 1.MD.4, organize, represent, and interpret data; and answer questions about the data points, as they interpret the data presented on a pictograph (conceptual understanding) and use it to solve compare/difference unknown word problems (application). Problem 1, “How many more rainy days than sunny days?” Students are provided with a weather pictograph showing sunny days, rainy days, and cloudy days.  

• Unit 9, Lesson 5, Exit Ticket, students engage with 1.NBT.4, add within 100 including a two-digit number and a one-digit number, using concrete models or drawings; understand that it is sometimes necessary to compose a ten, as they add a two-digit number by compose a ten (procedural skill), and explain how they solved the problem (conceptual understanding). Problem 1, “Solve. Show your work. 57 + 6 = ________. “ Problem 2, “How did you solve? Why?”

The materials reviewed for Achievement First Mathematics Grade 1 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 3, Lesson 14, Narrative engages students in making sense of a real-world situation. “Acting out will be a part of the Introduction today to provide support for students who may struggle to visualize these types of problems on their own immediately. Creating a representation that depicts what is happening in the story can be challenging, so teachers should be ready to support students with questions like ‘What do you know? What do you need to find out?’” Introduction, Step 1, “There were some butterflies in my net. Five of the butterflies escaped. There are still 7 butterflies in my net. How many butterflies were in my net to start?”

• Unit 7, Lesson 3, Understand: Introduce the Problem, “Pose the Problem- I’m going to read you a problem. As I read, I want you to make a mind movie just like we do in Math Stories to visualize what is happening and what we need to figure out. Dad bakes two small peach pies. Both small peach pies are the same size. Dad cuts one peach pie in halves. Dad cuts one peach pie in fourths. Dad says Max can eat just one piece of peach pie. Max loves peach pie and want to each the largest piece of peach pie. Which piece of pie does Max pick to eat? Show all of your mathematical thinking.”

• Unit 8, Lesson 11, Narrative encourages teachers to promote perseverance. “If students see that their representation does not match the story, they know that they have made an error and they need to represent differently before they solve.”

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

• Unit 3, Lesson 5, Introduction provides prompts to guide teachers that help students make sense of information. “What do you notice about these two number bonds showing addition and subtraction? They are the same parts and the same whole.” Later, in the Exit Ticket, Problem 1, students demonstrate how they made sense of the content. “Write an addition equation and a subtraction equation that match the number bond.” (Number bond with 11, 4, 7)

• Unit 5, Lesson 9, Narrative, “Students engage in MP 2 today when they represent a subtraction situation symbolically with equations and explain what each quantity represents (with part/part/whole understanding).” 

• Unit 8, Lesson 3, Narrative, “Students reason abstractly (MP 2) about the relative length of objects or characters in stories. They decontextualize the relative lengths by drawing a picture, creating a logical representation of the problem. Bob, Andy, and Joe are comparing their heights. Bob is taller than Andy. Bod is shorter than Joe. Who is the tallest?”

The materials reviewed for Achievement First Mathematics Grade 1 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 9, Exit Ticket, Problem 2, “Fallon put together her puzzle like this. She says she made a square. Is she correct? How do you know?” An image of a rhombus made up of four triangles is shown.

• Unit 3, Lesson 5, Introduction, students represent addition and subtraction scenarios with number bonds. The teacher asks, “How are addition and subtraction related? What’s the same about them? What’s different? How does the number bond show both addition and subtraction?” 

• Unit 3, Lesson 26, Introduction, students solve a story problem during the workshop introduction. “Ajacia saw 8 fewer red birds than brown birds in her yard. She saw 15 brown birds. How many red birds did she see?” The teacher looks for accurate representations and calls up students to explain their representations. The student might say, “I drew 15 circles for the 15 brown birds and put a box around it. Then I drew 8 x’s because I know that she saw 8 fewer red birds so she saw the same amount but 8 less. I put a box around that. I then put another box with a ? to represent that we need to figure out how many red birds that was.” 

• Unit 7, Lesson 1, Workshop Worksheet, students construct a viable argument for partitioning shapes into halves. “Draw a line to split each of these shapes into halves. How do you know they are half and half?” Students are provided with pictures of nine shapes: two triangles, one hexagon, two circles, one rectangle, two squares, and one oval.

• Unit 8 Assessment, students analyze the mathematical reasoning of others as they determine whether a fictional student measured the height of a tree correctly. Item 11, “Bobby says the tree is 4 toothpicks tall. Do you agree or disagree? Why?”

• Unit 9, Lesson 4, Share/Discussion, guiding questions are provided for teachers to lead students to analyze the reasoning of other students as they share how they added a two-digit number and a one-digit number. “2-3 students share their work/strategies (count on, make ten). What is the same about these strategies? What is different? Which strategy is more efficient?”

The materials reviewed for Achievement First Mathematics Grade 1 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 27, Introduction Task, “Hank grows 19 bean plants in his garden. A goat eats 6 of the bean plants. Hank buys 5 more bean plants to grow in his garden. How many bean plants does Hank now have in his garden? Show all of your mathematical thinking.” Narrative, “Today’s lesson also supports the development of MP5 as students have a large bank of tools and strategies to choose from when modeling and solving today. Students can use concrete objects, pictures, tape diagrams (1:1 or numerical), number bonds (1:1 or numerical), and /or equations as strategies/tools to model the problem.”

• Unit 6, Lesson 4, Narrative, students decompose numbers 10-99 into tens and ones by using cubes, pictures, or knowledge of place value. “Students will look at a number, decompose into tens and ones, and represent using literal pictures, sticks and dots or an equation.” 

• Unit 9, Lesson 8, Introduction Step 2, “Yesterday we used cubes to help us solve. But today we won’t all have cubes….Could we show this with a picture? How could we show regrouping with a picture?”

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

• Unit 1, Lesson 9, Narrative, “Students engage in MP5 when they choose an appropriate strategy to determine the number that comes right before/after a number or between two numbers. Students may access the number line or hundreds chart as useful classroom tools that are always available to them, but it is expected that most students will use strategy such as using the pattern to count from a benchmark number or to just know the next number and some students might use place value understanding. All of these strategies and tools are highlighted and discussed as they come up.”

• Unit 3, Lesson 7, Exit Slip Question 1, “Solve for the unknown. Write an addition equation that shows the parts and whole. (You may use the number line but do not have to.)” The problem includes a number bond with 8 as the whole, 5 as one part, and one part blank. The problem also includes a number line. 

• Unit 9, Lesson 10, Narrative, “Students also develop MP 5 as they choose from a variety of tools and strategies to add two-digit numbers today. They may use cubes, pictures, fingers, or expanded notation models to help them solve. They can count all, count on in increments, or use known facts. Students explain why they chose the tools/ strategies they did both orally through CFUs (check for understanding) in the workshop and in writing on their workshop packets.”

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 1 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 3, Lesson 4, Exit Slip Question 1, students use precision to find the difference between two numbers and write an equation to match the problem. “Find the difference between the number cubes. Represent by completing the number bond and the equation.”

• Unit 5, Lesson 4, Skeleton VA, students use precision to prove the associative property. “Associative Property: what happens when we are combining amounts and we group the amounts differently. We get the same total! When I add, it doesn’t matter how I group the numbers--the total is the same.” Students are provided with pictures of three dice showing four, three, and two dots, and group the numbers in three different ways to demonstrate that they always have nine dots in all.

• Unit 6, Lesson 6, Criteria for Success, students use precision to show place value. ‘Students will roll two dice to determine the digits in the tens place and ones place. Then they will need to figure out how many by using their understanding of place value. They will also need to represent the quantity with literal pictures or sticks and dots, and expanded notation. Scholars should be able to explain their representation by saying, “This number is 62. I know because there is a 6 in the tens place which means there are 6 groups of ten, so I drew ten sticks. There is a 2 in the ones place which means there are 2 ones so I drew 2 dots to show the 2 ones. Then I wrote 60 + 2 to show the value of the tens and ones.”

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

• Unit 2, Lesson 10, Introduction, Step 3, “Describe the shape. Take a minute and look closely at the shape. What are some attributes we notice about this cone? (as students share, generalize to bullets below- ‘Oh, you noticed that it rolls, so you thought about it MOVES!’ Be sure to use appropriate language for attributes and prompt kids to do so as well: ‘That’s called a vertex; say it again but this time say “vertex” instead of point.’”

• Unit 7, Lesson 2, Introduction, during a game the teacher develops vocabulary. “Step 1 says Look at the shape. Step 2 says Break it in Quarters. How can I break this rectangle in quarters?’ Students might say, ‘You should draw a line down the middle so that it’s in two equal parts. Then draw another line down the middle so it’s four equal parts.’”

• Unit 8, Lesson 9, the Criteria for Success has an exemplar student response using accurate terminology. “The knife is 2 inch tiles longer than the fork. I know because I measured each with inch tiles, match the inch tiles one to one, and I could see that the knife needed 2 extra inch tiles to measure it, so it was 2 inch tiles longer.”

The materials reviewed for Achievement First Mathematics Grade 1 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 1, Lesson 8, Narrative, “Students engage with MP 7 today as they observe patterns in how we say and write numbers and apply those patterns to extend the sequence. Teachers help students to see more patterns today by using the structure of the hundreds chart. As noted above, the arrangement of the numbers into rows of ten helps students to see more clearly the pattern in the tens place. Students may also begin to recognize the structure of place value.”

• Unit 5, Lesson 5, Assessment and Criteria for Success, “Student solves to find the totals 18 and 13. Student shows how he/she grouped the numbers for both problems and how they made ten for number 2.” (Problem posed: 7 + 6 + 5)

• Unit 7, Lesson 6, Workshop Worksheet, students use their understanding of the structure of a circle and fractions to tell time to the nearest half hour. Question 3, “Draw the missing hands on the clock. Half past 4.” Students are provided a picture of a clock with no hands.

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

• Unit 2, Lesson 3, Narrative, “They also engage with MP 8 as they use repeated reasoning to determine the ‘rule’ their partner is using to sort the shapes.” 

• Unit 6, Lesson 2, Introduce the math, “Yesterday we got to compose numbers by counting by tens and ones, and drawing pictures to show how many. Today we’re going to use what we know to do the same thing, but with even BIGGER numbers! What do we notice about how we write the digits of numbers and how that relates to the number of towers and extra cubes?”

Unit 9, Lesson 2, Exit Ticket, students use regular repeated reasoning to relate what they have learned about adding a two-digit number and a multiple of ten using sticks and dots, to adding them using the strategy of expanded notation. Problem 2, “Solve using expanded notation. 37 + 50 = ____.”