The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations for assessing grade-level content. Summative Interim Assessments include Beginning-of-Year, Mid-Year, and End-of-Year. Above-grade-level assessment items are present but could be modified or omitted without a significant impact on the underlying structure of the instructional materials.

Examples of aligned assessment items include but are not limited to:

• Unit 3 Open Response Assessment, Item 1, “Use the number line to help you solve the story. You are collecting leaves. You have 3 leaves in your pocket. You pick up some more leaves. Now you have 10 leaves. How many leaves did you pick up?” (1.OA.1)
• Unit 4 Assessment, Item 6, “Ali has 7 red crayons, 3 yellow crayons, and 7 blue crayons. How many crayons does he have in all? Explain how you found the sum.” (1.OA.2)
• Unit 6 Cumulative Assessment, Item 2, “Alice says that if she knows that 8 + 9 = 17, then she also knows that 9 + 8 = 17. Is Alice correct? Explain why or why not.” (1.OA.3)
• Mid-Year Assessment, Item 13, “Shelby and James used paper clips to measure a marker. Shelby measured like this: James measured like this: Who measured correctly? Tell why you think so.” (1.MD.2)

There are some above-grade-level assessment items that can be omitted or modified. These include: 

• Unit 3 Assessment, Item 7, “Fill in the rule and the frames.” (4.OA.5)
• Unit 7 Assessment, Item 11, “Find the rule. Fill in the missing numbers.” Students look at a function table containing in and out boxes, determine the rule, and fill in the missing numbers. (4.OA.5)
• End-the-Year Assessment, Item 5, “Fill in the rule and the missing numbers.” Students find the pattern, subtract 10, and fill in the missing numbers and rule when shown 95, 85, 75, ___, ___, 45, 35. (4.OA.5)

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations for spending a majority of instructional time on major work of the grade.

• There are 9 instructional units, of which 6 units address major work of the grade or supporting work connected to major work of the grade, approximately 67%.
• There are 109 lessons, of which 72.5 address major work of the grade or supporting work connected to the major work of the grade, approximately 67%.
• In total, there are 170 days of instruction (109 lessons, 37 flex days, and 24 days for assessment), of which 101 days address major work of the grade or supporting work connected to the major work of the grade, approximately 59%. 
• Within the 37 Flex days, the percentage of major work or supporting work connected to major work could not be calculated because the materials suggested list of differentiated activities do not include explicit instructions. Therefore, it cannot be determined if all students would be working on major work of the grade.

The number of lessons devoted to major work is most representative of the instructional materials. As a result, approximately 67% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Examples of supporting standards/clusters connected to the major standards/clusters of the grade include but are not limited to:

• In Lesson 1-1, Focus: Estimating and Counting Collections of Objects, students guess the number of crayons a teacher has in a bag. After students make estimations, they count the actual number of crayons and determine the reasonableness of their guess. This connects supporting standard 1.G.1, “Distinguish between defining attributes versus non-defining attributes,” to the major work of 1.NBT.1, “Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.” 
• In Lesson 1-8, Focus: Organizing and Representing Data in a Tally Chart, students count and tally topics such as how children get to school, their favorite recess activity, or type of bottom they are wearing given 3 categories. This data is then represented in tally tables and used to answer questions requiring data interpretation. This connects the supporting standard 1.MD.4, “Organize, represent, and interpret data with up to three categories,” to the major work of 1.OA.6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.”
• In Lesson 4-6, Focus: Building a Superhero Bar Graph, students make a tally chart of what superhero power classmates would choose (to fly, be invisible, or have extra strength). Students then make a bar graph of the same data and answer questions comparing the data. This connects the supporting standard 1.MD.4, “Organize, represent, and interpret data with up to three categories,” to the major work of 1.OA 6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.”
• In Lesson 7-11, Focus: Introducing the Minute Hand, students discuss the length of a minute and an hour, while identifying the minute and second hand on an analog clock. Students clap in unison and count up to 60 to know the length of a minute. This connects the supporting standard 1.MD.3, “Tell and write time in hours and half-hours using analog and digital clocks,” to the major work of 1.NBT.1, “Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.”
• In Lesson 8-6, Focus: Making a Shapes Bar Graph, students practice building composite shapes with square, triangle, and trapezoid pattern blocks. Students then record the number of shapes they used on a bar graph and work with a partner to ask questions about the graphs. This connects the supporting work of 1.G.2, “Compose two-dimensional shapes or three-dimensional shapes to create a composite shape,” to the major work of 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.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations that the amount of content designated for one grade level is viable for one year. 

Recommended pacing information is found on page xxii of the Teacher’s Lesson Guide and online in the Instructional Pacing Recommendations. As designed, the instructional materials can be completed in 170 days:

• There are 9 instructional units with 109 lessons. Open Response/Reengagement lessons require 2 days of instruction adding 9 additional lesson days.
• There are 37 Flex Days that can be used for lesson extension, differentiation, games, etc; however, explicit teacher instructions are not provided.
• There are 24 days for assessment which include Progress Checks, Open Response Lessons, Beginning-of-Year Assessment, Mid-Year Assessment, and End-of-Year Assessment.   

The materials note lessons are 60-75 minutes and consist of 3 components: Warm-Up: 10-15 minutes; Core Activity: Focus: 30-35 minutes; and Core Activity: Practice: 15-20 minutes.

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations for being consistent with the progressions in the Standards. The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades and with work in future grades. The instructional materials also present extensive work with grade-level problems.

The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Each Section Organizer contains a Coherence section with “Links to the Past”. This section describes “how standards addressed in the Focus parts of the lessons link to the mathematics that children have done in the past.” Examples include:

• Teacher’s Lesson Guide, Section 1 Organizer, Coherence, “Links to the Past” for 1.NBT.1, “In Kindergarten, children learned to count to 100 by 1s and by 10s.”
• Teacher’s Lesson Guide, Section 4 Organizer, Coherence, “Links to the Past” for 1.MD.2, “In kindergarten, children identified measurable attributes of objects, including length. They also used direct comparison to determine which of two objects is longer.”
• Teacher’s Lesson Guide, Section 5 Organizer, Coherence, “Links to the Past” for 1.NBT.4, “In Kindergarten, children developed an understanding of teen numbers as 10 ones and some further ones.”
• Teacher’s Lesson Guide, Section 6 Organizer, Coherence, “Links to the Past” for 1.OA.3, “In Unit 2, children developed the Turn-Around Rule as a strategy to more efficiently solve addition facts. The flexibility needed to make sense of this strategy was first developed in Kindergarten as children learned to decompose numbers less than or equal to 10 in different ways.”
• Teacher’s Lesson Guide, Section 8 Organizer, Coherence, “Links to the Past” for 1.G.2, “In Unit 7, children explored 2-dimensional shapes with different attributes, reviewing various common shapes such as triangles and rectangles. In Kindergarten, children composed small shapes to form larger shapes.”

The instructional materials relate grade-level concepts with work in future grades. Each Section Organizer contains a Coherence section with “Links to the Future”. This section identifies what students “will do in the future.” Examples include:

• Teacher’s Lesson Guide, Section 1 Organizer, Coherence, “Links to the Future” for 1.OA.5, “In Unit 3, children will utilize number lines to keep track of counts as they solve addition and subtraction problems. In Grade 2, children will use their understanding of the relationship between addition and counting to make sense of the properties of even and odd numbers.”
• Teacher’s Lesson Guide, Section 3 Organizer, Coherence, “Links to the Future” for 1.NBT.1, “In Unit 5, children will use the patterns they observed when counting within 100 to expand the number grid to larger numbers as they create number scrolls. In Grade 2, children will extend this even further as they count and represent numbers to 1000.”
• Teacher’s Lesson Guide, Section 4 Organizer, Coherence, “Links to the Future” for 1.MD.1, “In Unit 4, children will progress from direct comparisons of length to indirect comparisons of length using a third object. Then in Unit 5, they will use indirect comparisons to order heights of fixed objects that can’t be measured directly. In Grade 2, children will use standard units to express the difference in lengths of two objects.”
• Teacher’s Lesson Guide, Section 6 Organizer, Coherence, “Links to the Future” for 1.NBT.4, “In Unit 9, children will revisit adding and subtracting within 100. In Grade 2, children will extend their strategies to solve addition and subtraction problems within 1000, with a focus on developing fluency within 100.”
• Teacher’s Lesson Guide, Section 8 Organizer, Coherence, “Links to the Future” for 1.G.3, “In Unit 9, children will review partitioning into halves and quarters and continue to reason why decomposing into more equal shares creates smaller shares. They will build upon those ideas in Grade 2 as they learn to partition a whole into thirds.”

Lesson 3-5 contains content from future grades that is not clearly identified as such. In Focus: Reviewing Skip Counting on Number Lines, “When children have completed the journal page, encourage them to discuss and compare any patterns they see in the different skip-counts. Ask: Why does it take more hops to count to 20 by 5s than it does by 10s?” This lesson is labeled 1.NBT.1, “Counting to 120, starting at any number less than 120.” Counting by 5s and 10s is a Grade 2 standard (2.NBT.2, “Count within 1,000; skip-count by 5s, 10s, and 100s”).

The instructional materials give students extensive work with grade-level problems except for 1.NBT.1, “Count to 120, staring at any number less than 120.” This standard is addressed in the Focus section of several lessons throughout Units 1-5 and embedded in Daily Routines which explore and extend real-world application of math. However, 2 of these lessons, Lesson 1-11 and 3-8, involve number charts to 120. In addition, examples given to teachers were within 100 (Lesson 1-11, Focus: Introducing the number grid, the materials provide a total of 12 examples for students to practice counting up and counting back by 1s and 10s. All 12 examples are within 80. For example, “Start at 26 and count back 4 hops. Where do you land?” or “Start at 70 and count back 10 hops. Where do you land?”)

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards.

Materials include learning objectives that are visibly shaped by CCSSM cluster headings. Focus and Supporting Clusters addressed in each section are found in the Table of Contents, the Focus portion of each Section Organizer, and in the Focus portion of each lesson. Examples include:

• Lesson 1-9, Focus: Exploring with Pattern Blocks, Base-10 Blocks, and Geoboards is shaped by 1.NBT.A, “Extend the counting sequence.” Students use pattern blocks to create designs and patterns, base-10 blocks to build structures, and geoboards to make various shapes and pictures.
• Lesson 3-6, Focus: Introducing Addition on the Number Line is shaped by 1.OA.A, “Represent and solve problems involving addition and subtraction.” Students use a number line to solve addition and subtraction problems.
• Lesson 4-6, Focus: Building a Superhero Bar Graph is shaped by 1.MD.C, “Represent and interpret data.” Students create a bar graph representing which superpower classmates choose (to fly, to be invisible, or to be extra strong).
• Lesson 8-1, Focus: Constructing Straw Polygons is shaped by 1.G.A, “Reason with shapes and their attributes.” Students review defining attributes and create polygons using straws.
• Lesson 9-9, Focus: Reviewing Place Value is shaped by 1.NBT.B, “Understand place value.” Students use base-10 blocks to demonstrate 2-digit numbers.

The materials include problems and activities connecting two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. Examples include:

• In Lesson 1-5, Focus: Bunny Hop, while playing a game, students use a number line to count up and back. 1.OA.C, “Add and subtract within 20 connects” to 1.NBT.A, “Extend the counting sequence.”
• In Lesson 2-10, Focus: Introducing Addition Number Models, students “use plus and equal signs to write number models for change-to-more word problems.” 1.OA.A, “Represent and solve problems involving addition and subtraction” connects to 1.OA.C, “Add and subtract within 20.”
• In Lesson 3-6, Focus: Introducing Addition on the Number Line, students use a number line to solve addition problems and share their solution strategies. 1.NBT.A, “Extend the counting sequence” connects to 1.OA.C, “Add and subtract within 20.”
• In Lesson 4-8, Focus: Fact Strategy Review, students use strategies to solve addition facts and record the facts they know. 1.OA.B, “Understand and apply properties of operations and the relationship between addition and subtraction” connects to 1.OA.C, “Add and subtract within 20.”
• In Lesson 5-7, Focus: Measuring a Path, students measure the length of a crooked path with paperclips. 1.MD.A, “Measure lengths indirectly and by iterating length units” connects to 1.OA.C, “Add and subtract within 20.”
• In Lesson 5-11, Focus: Subtracting Animal Weights, students use a variety of strategies to find the difference in weights of pairs of animals. 1.NBT.B, “Understand place value” connects to 1.NBT.C, “Use place value understanding and properties of operations to add and subtract.”
• In Lesson 8-8, Focus: Introducing Time to the Half Hour, students shade a clock face and determine how much time this represents. 1.MD.B, “Tell and write time” connects with 1.G.A, “Reason with shapes and their attributes.”
• In Lesson 8-10, Focus: Reviewing Place Value on the Number Grid, students review place-value patterns shown on the number grid and model the patterns with base-10 blocks. 1.NBT.B, “Understand place value” connects to 1.NBT.C, “Use place value understanding and properties of operations to add and subtract.”
• In Lesson 9-5, Focus: Adding 2-Digit Vending Machine Prices, students use different strategies to add prices of items from a vending machine. 1.OA.A, “Represent and solve problems involving addition and subtraction” connects to 1.NBT.C, “Use place value understanding and properties of operations to add and subtract.” 

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade level. The Focus portion of the lesson provides opportunities for students to explore, engage in, and discuss conceptual understanding of mathematical content. Examples include:

• In Lesson 1-6, Focus: Comparing and Ordering Numbers, students use number cards from 1-15 to compare numbers and to order sets of numbers. Students mix their cards up, draw two cards, and use the class number line to decide which number is larger. Students are encouraged to, “Use comparative language such as ‘8 is larger than 2, and 12 is smaller than 15.’” This activity supports conceptual understanding of 1.NBT.3, “Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.”
• In Lesson 3-1, Focus: Introducing Domino Addition, students represent the number of dots on dominoes with parts-and-total diagrams. For example, for a domino with 3 and 5 dots students write Part: 3, Part: 5, and Total: 8 on the diagram in their math journal and then write the corresponding number sentence. They use the diagrams to record the corresponding number sentences. This activity supports conceptual understanding of 1.OA.6, “Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing unknowns in all positions.”
• In Lesson 5-1, Focus: Exchanging Base-10 Blocks, “Display 1 long and 15 cubes on your mat. Ask: What number is shown? Discuss how children can be sure. Demonstrate the following two exchanges, or trades, starting with 1 long and 15 cubes each time: Trade the long for 10 cubes, then count the total number of cubes to get to 25. Trade 10 cubes for 1 long and place the long in the tens column. As you trade, emphasize that you are making another ten (1 long) from 10 ones (10 cubes). There are now 2 longs and 5 cubes. Ask: What number is shown? What does the 2 represent? What does the 5 represent?” This activity supports conceptual understanding of 1.NBT.2, “Understand that the two digits of a two-digit number represent amounts of tens and ones.” 
• In Lesson 7-6, Focus: Making an Attribute Train, one student brings an attribute block to the front of the room and is designated the conductor. They choose a child to join the train and that student must bring a block to add to the train that differs from the conductor’s block in only one way. This activity supports conceptual understanding of 1.G.1, “Distinguish between defining attributes versus non-defining attributes, build and draw shapes to possess defining attributes.”
• In Lesson 8-11, Focus: Adding and Subtracting 10 Mentally, students apply strategies for adding or subtracting 10. Teachers are prompted, “Discuss how children found their answers to the Math Message (35 cents). Emphasize strategies, including visualizing a number grid and moving down a row from 25 to 35, thinking about adding a long to base-10 blocks representing 25, and adding 1 to the tens digit of 25.” This activity supports conceptual understanding of 1.NBT.5, “Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count to explain the reasoning used.”

Games, Daily Routines, and Math Journals provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include:

• In Routine 1: Number of the Day, each day a student adds a straw to the chart. On day 10, a student is chosen to take the 10 straws out of the ones cup, bundle them with a rubber band, and place them in the 10s cup. Students continue bundling groups of 10 straws and on Day 100 they remove all 10 bundles from the cup, bundle them with a rubber band and place them in the hundreds cup. This activity provides continuous conceptual understanding practice of 1.NBT.2, “Understand that the two digits of a two-digit number represent amounts of tens and ones.”
• In Lesson 5-8, Game: Base-10 Exchange, using a Ten-And-Ones Mat (Activity Sheet 4), base-10 blocks, and one dot die, students play with a partner practicing exchanging ones for tens after rolling the die to determine how many base-10 blocks to get. The first student to get 10 longs wins. Teachers are prompted, “As children play, be sure to ask questions that reinforce place-value concepts.” This activity provides practice of conceptual understanding of 1.NBT.2, “Understand that the two digits of a two-digit number represent amounts of tens and ones.”
• In Lesson 4-3, Math Journal, students independently practice measuring objects such as their desktop using one new pencil as the unit. Then students draw a picture or write the name of the measured object and record its measurement. This activity supports the conceptual understanding of 1.MD.2, “Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object 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.”
• In Lesson 8-2, Math Journal, students independently partition pictures of pancakes and crackers in equal shares and explain how they know the share is equal. Question 2 states, “Show how to share 1 cracker between 2 people.” This activity supports the conceptual understanding of 1.G.3, “Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and a quarter of.”
• In Lesson 9-2, Math Journal, students record their own number stories using items from the School Store Mini-Poster that lists items and their prices. Sample story states, “I bought a bookmark and an eraser. I paid 43¢ in all.” Use the Number Model: 26¢ + 17¢ = 43¢. This activity supports the conceptual understanding of 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 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. The Section Organizer provides information on which part of each lesson develops procedural skill and fluency. Opportunities are found in the Daily Routines and Focus portions of the lesson. Examples include:

• In Lesson 2-1, Focus: Introducing Roll and Total, students take turns with a partner to roll one numeral die and one dot die and start with the numeral die to count on by the number of dots and record their sums. Students should recognize that it is more efficient to count on from the larger number being added. This activity provides an opportunity for students to develop fluency of 1.OA.5, “Relate counting to addition and subtraction.”
• In Lesson 3-3, Focus: Counting Large Numbers of Pennies, groups of students are given a container of at least 50 pennies. They estimate the total, then count and record the number of pennies. This activity provides an opportunity for students to develop fluency of 1.NBT.1, “Count to 120, starting at any number less than 120.”
• In Lesson 3-6, Focus: Introducing Addition on the Number Line, students represent and make sense of word problems by drawing hops on a number line. Students answer questions such as, “Cynthia had 8 model cars. She got 3 more model cars. How many model cars does Cynthia have now?” This activity provides an opportunity for students to develop fluency of 1.OA.6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.”
• In Lesson 4-10, Focus: Adding Three Numbers, teachers support students in learning the procedure of adding three numbers by finding combinations of 10 and doubles. For example, when adding 4 + 3 + 7, students can “Add 4 and 3 first to make 7; then double 7.” The teacher gives students additional problems to solve. Teachers are told to “emphasize that grouping numbers to make combinations of 10 or using doubles makes adding easier. Discuss some problems that do not contain a combination of 10 or doubles (for example, 7 + 9 + 4). Guide children to understand that when three or more numbers are added, it does not matter which two are added first; the sum is still the same.” This activity provides an opportunity for students to develop fluency of 1.OA.2, “Solve word problems that call for the addition of three whole numbers whose sum is less than or equal to 20.” 
• In Routine 2: Calendar Routine, students use the calendar to answer daily questions such as, “Find the sum of two dates on the calendar. Find the number of days from today’s date until another date.” This activity provides a continuous opportunity for students to develop fluency of 1.OA.6, “Add and subtract within 20 demonstrating fluency for addition and subtraction within 10.”
• In Routine 6: Math Any Time Routine, the class daily schedule can be used to practice time concepts. Students are given their own clocks and asked questions such as, “How long does lunch last? It is 9:00 a.m. now, how long will it be until we start math? Which activity takes the shortest amount of time today?” This activity provides a continuous opportunity for students to develop fluency of 1.MD.3, “Tell and write time in hours and half-hours using analog and digital clocks.”

The instructional materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. The Section Organizer provides information on which part of each lesson develops procedural skill and fluency. Opportunities are found in the Practice portions of the lesson, Math Journal, and Math Masters. Examples include:

• In Lesson 2-3, Math Journal 1, Question 1, “Count up by 1s. 7, 8, 9,____, ____, ____, ___, ____, ____, ____, ___”. This provides an opportunity for students to independently demonstrate the procedural skill of 1.NBT.1, “Count to 120 starting at any number less than 120.”
• In Lesson 6-5, Math Journal 2, students work with a partner to use the strategy of near doubles to solve story problems. The story states, “I have 5 pencils in my desk and 7 pencils in my backpack. How many more pencils do I have all together?” A possible response is, “I know 7 + 7 is 14, so 7 + 5 is 2 less, or 12. So, 7 + 7 - 2 = 12.” This activity provides an opportunity for students to independently demonstrate the procedural skill of 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.”
• In Lesson 8-1, Math Journal 2, students write fact families from fact triangles (11, 6, and 5) dominos (8 and 6). In Question 4, “How can 9 + 3 = 12 help you solve 12 - 9?” This activity provides an opportunity for students to independently demonstrate the procedural skill of 1.OA.6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.”
• In Lesson 8-2, Math Journal, Question 4, “Use your number grid. Add. 17 + 10 = ____, 17 + 20 = _____, 17 + 30 = ____”. This activity provides an opportunity for students to independently demonstrate the procedural skill of 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 10.”
• In Lesson 9-4, Math Journal, students complete a “My Facts Inventory Record”. Students see addition facts such as 10 + 2 and check a box if they know it or don’t know it, then explain how they can figure it out. This activity provides an opportunity for students to independently demonstrate the procedural skill of 1.OA.6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. The materials do not provide opportunities for students to independently engage in non-routine applications of mathematics throughout the grade level.

Examples of students engaging in routine application of mathematics include:

• In Lesson 3-4, Focus: Representing a Number Story, students use a picture and a model to represent a situation, “There were 9 birds sitting in a tree. Some birds flew away. 5 birds stayed. How many birds flew away?” Students apply their understanding of 1.OA.8, “Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.”
• In Lesson 4-10, Focus: Adding Three Numbers, students add three numbers from School Supply Cards. Teachers tell the following number story, “Our class has 7 pencils, 4 pens, and 3 crayons. How many writing tools do we have in all?” Students independently solve using number models and discuss their strategies with other students. Students apply their understanding of 1.OA.2, “Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20.”
• In Lesson 5-10, Focus: Finding How Much More or Less, students solve comparison number stories such as, “Alberto has 12 cents. June has 7 cents. Who has more money? How much more money?” Students are encouraged to use comparison diagrams to solve. Students apply their understanding of 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.”
• In Lesson 6-2, Focus: Interpreting and Solving Animal Number Stories, students use information on an Animal Card containing heights/lengths and weights to solve number stories such as, “A cat and a raccoon sit on a scale at the same time. How much do they weigh together?” Students are encouraged to model the situation prior to writing their number model. Students apply their understanding of 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.”

The materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Examples include:

• In Lesson 3-2, Practice: Modeling Number Stories, students write number models to solve stories such as, “Walt was at the carnival. He had 8 carnival tickets and 2 pens. He traded 4 tickets for 1 more pen. How many tickets does Walt have now? How many pens does Walt have now?” This activity provides the opportunity for students to independently demonstrate 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.”
• In Lesson 5-3, Practice: Comparing Similar Lengths, students compare the lengths of several objects and answer questions such as, “Which is longer, the pencil or the pen? Which is shorter, the toothbrush or the crayon?” This activity provides the opportunity for students to independently demonstrate 1.MD.1, “Order three objects by length; compare the lengths of two objects indirectly by using a third object.”
• In Lesson 7-6, Practice: Solving Garden Number Stories, students practice solving number stories such as, “In the garden, there are 6 green peppers, 7 yellow squash, and 4 yellow peppers. How many vegetables are in the garden?” This activity provides the opportunity for students to independently demonstrate 1.OA.2, “Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20.”
• In Lesson 8-4, Focus: Solving the Open Response Problem, students use a drawing to determine which is a larger square when solving, “In art class, children sit at tables for a project. Two girls sit at one table and four boys sit at another table. Each table gets one large square of construction paper. The two girls share their paper square evenly, and the four boys share their paper square evenly. Who gets a larger share of paper, one girl or one boy?” This activity provides the opportunity for students to independently demonstrate 1.G.3, “Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrase half of, fourth of, and a quarter of.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present independently throughout the program materials. 

The materials attend to conceptual understanding. Examples include:

• In Lesson 2-9, Focus: Solving Mystery Cup Problems, students solve mystery cup-drop problems, “Explain that children will work in partnerships to make sense of and solve cube-knocking mysteries. Explain that they will determine what information is missing. For example, pose the following problem: Imagine there are 10 cups. I knock some over. There are 7 cups left standing. How many cups did I knock over?” This activity develops conceptual understanding of 1.OA.4, “Understand subtraction as an unknown-addend problem.”
• In Lesson 5-2, Focus: Investigating Base-10 Block Patterns, students make place-value exchanges and discuss various ways to represent numbers, “Tell children that today they will explore patterns in numbers using both base-10 blocks and calculators. Display the following with your Tens-and-Ones Mat as children follow along on their own mats. 1. Display 9 cubes in the ones column. 2. Add 1 cube to the ones column. 3. Exchange the 10 cubes for 1 long and put it in the tens column. This develops conceptual understanding of 1.NBT.2, “Understand that the two digits of a two-digit number represent amounts of tens and ones.”

The materials attend to procedural skill and fluency. Examples include:

• In Lesson 4-8, Focus: Fact Strategy Review, students discuss addition facts and note which facts they know well on a Facts Inventory Record, “Explain that addition facts are two numbers from 0 to 10 and their sum. Point out that children can use the strategies on the Strategy Wall to solve addition facts. Record children’s ideas about adding 0 on the Strategy Wall.” This activity develops the procedural skill of 1.OA.6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.”
• In Lesson 9-3, Focus: Solving the Open Response Problem, students choose three items to buy, find the total cost, and show or explain their strategies, “Tell children that they will work in pairs. Each child should choose three items from the poster and find the total amount needed to pay for the items, recording his or her work and answer on the Math Masters page. Then partners take turns explaining to each other how they solved the problem. Encourage partners to ask questions if they do not understand an explanation. This activity develops the procedural skill of 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 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”

The materials attend to application. Examples include: 

• In Lesson 3-8, Focus: Using Number-Grid Counting to Add and Subtract, students solve number stories by skip counting on a number grid, “Tell children the following number story: Justin’s teacher gives out stickers for good behavior. At the beginning of the day, Justin had 0 stickers. During the day, he earned 10 stickers. How many stickers does Justin have at the end of the day? Have children share solutions to the problem.” This activity provides the opportunity to apply understanding of 1.OA.3, “Apply properties of operations as strategies to add and subtract.”
• In Lesson 6-5, Focus: Using Near Doubles to Solve Stories, students solve a number story and explain their strategy with pictures, words or symbols, “Tell children that they will now practice recognizing when to use the near-doubles strategy to solve number stories. Remind them that near means close to a double. Present the following number story: I have 5 pencils in my desk and 7 pencils in my backpack. How many pencils do I have all together? Ask children to solve this problem using two different helper facts.” This activity provides the opportunity to apply understanding of 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.”

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

• In Lesson 2-1, Focus: Introducing the Turn-Around Rule, students discover and define the turn-around rule for addition, solving 2 problems, “Solve. Show your work. Use drawings, numbers, or words. 1. On Monday, Ellie found 2 pennies. On Tuesday, Ellie found 3 pennies. How many pennies did Ellie find in all? 2. On Monday, Tommy found 3 pennies. On Tuesday, Tommy found 2 pennies. How many pennies did Tommy find in all?” After solving, students discuss how they solved with a partner. “Help children recognize that both problems involve putting together two parts, which is called adding. They should also observe that both problems have the same numbers and the same total, or sum, but the order of the numbers being added is different.” Students develop procedural skill of 1.OA.6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10,” and application of 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.”
• In Lesson 6-6, Focus: Math Message and Developing the Making-10 Strategy, students first show how they represent numbers on a double ten frame, then they represent making-10 strategy, “How could you show 15 on a double ten frame? How about 18? 19? Record your thinking on your slate. Display a double ten frame. Have children use their own double ten frames and counters to share their thinking. Although children may represent the numbers in the Math Message in a variety of ways, emphasize filling one ten frame with ten and then repeating the remaining ones on the second ten frame. Generalize this idea by asking: How can you represent teen numbers?” Students develop conceptual understanding of 1.NBT.2, “Understand that two digits of a two-digit number represent amounts of tens and ones,” and procedural skill with 1.NBT.1, “Count to 120, starting at any number less than 120.”
• In Lesson 8-8, Focus: Math Message and Introducing Time to the Half Hour, students are introduced to half-past an hour and shade half of a clock face, “Draw the clock face, and shade half of the clock. How much time has passed if the minute hand begins at 12 and goes through all of the shaded part? Children share their drawings. Compare the different representations and discuss how both pieces of the clock must be equal to be divided in half. Ask: What would you name the part of the clock that you shaded? Tell children that today they will use what they know about halves to learn more about telling time.” Students develop conceptual understanding of 1.G.3, “Partition circles and rectangles into two and four equal shares,” and procedural skill with 1.MD.3, “Tell and write time in hours and half-hours using analog and digital clocks.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade level.

All MPs are clearly identified throughout the materials, with few or no exceptions. Examples include:

• The mathematical practices are listed on pages xxxvi-xxxix in the Grade 1, Volume I, Teacher’s Lesson Guide as “Correlation to the Mathematical Process and Practices” which states, “Everyday Mathematics is a standards-based curriculum engineered to focus on specific mathematical content, processes, and practices in every lesson and activity. The chart below shows complete coverage of each mathematical process and practice in the core program throughout the grade level.” 
• Each Unit Organizer contains a Mathematical Background: Processes and Practices component identifying the MPs addressed in the section and in individual lessons. Additionally, “The authors created Goals for Mathematical Practice (GMP) that unpack the practice standards, operationalizing them in ways that are appropriate for elementary students.” 
• Within each lesson description, GMPs appear in bold print and teacher side notes identify the MPs that are addressed in the lesson.

The majority of the time the MPs are used to enrich the mathematical content. Examples include:

• In Lesson 2-11, Focus: Modeling Unknowns, students write number models for number stories, “Remind children about their recent work with number models. Tell them that today they will write more number models to represent number stories. Jackie dropped 3 pennies in a cup. Then she dropped 4 more pennies in the cup. She dropped 7 pennies in all. Display a change-to-more diagram and have children help you complete it. Have children suggest a representative number model and record it.” The mathematical content in this activity is enriched by MP4.
• In Lesson 3-6, Focus: Introducing Addition on the Number Line, students use the number line to solve addition problems and share their solution strategies, “Next ask children to draw hops on the third number line to solve 2 + 9. Select children to show counting on from 2 and counting on from 9. Discuss which strategy is more efficient. Help children generalize that counting on from the larger number, or 9, is faster because there are fewer numbers to count.” The mathematical content in this activity is enriched by MP8.
• In Lesson 5-8, Exploration A: Introducing Base-10 Exchange, students explore the relationship between tens and ones by playing, “As children play, be sure to ask questions that reinforce place-value concepts. When base-10 blocks represent teen numbers, ask: What number is shown by your cubes? How do you write that as a numeral? What is the value of the tens digit? What is the value of the ones digit?” The mathematical content in this activity is enriched by MP2.
• In Lesson 7-3, Focus: Think Addition to Subtract with Combinations of 10, students use addition to solve subtraction problems, “Point out that understanding combinations of 10 can also be useful for games such as Fishing for 10. For example, if they are holding a 3, they may think 3 + __ = 10 or 10 - 3 = __. Therefore, thinking of addition might help them solve problems involving decomposing a 10.” The mathematical content in this activity is enriched by MP7.
• In Lesson 8-9, Focus: Math Message, the class is shown a tally chart with some missing tallies for one of the 3 categories, “Mr. Chan’s class took a survey to figure out which pet is the favorite. They made a tally chart, but the tallies for Turtle got erased. If 19 children voted, how many voted for Turtle? What do you need to do to solve this problem? Record your answer, and explain how you found it.” The mathematical content in this activity is enriched by MP1.

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 partially meet expectations for carefully attending to the full meaning of each practice standard. The materials attend to the full meaning of most of the MPs, but they do not attend to the full meaning of MP5 as students do not get to choose tools strategically.

Examples of the materials attending to the full meaning of most MPs include:

• MP1: In Lesson 2-8, Focus: Introducing Change-to-More Diagrams, students use change-to-more diagrams to practice adding, “List some of the children’s predictions and have them explain their thinking. Ask the class whether the predictions seem reasonable. For example, predicting a sum that is smaller than one of the numbers being added would not be reasonable.” 
• MP2: In Lesson 5-1, Focus: Naming Numbers with Base-10 Blocks, students use base-10 blocks to represent numbers, “Have children find the total number of cubes. Place 3 longs and 4 cubes on the mat. How many cubes are shown? How do you know?”
• MP4: In Lesson 9-2, Focus: Recording School Store Number Stories, students write and solve number stories using the School Store Mini-Posters, “Tell different types of stories using items from the School Store Mini-Posters on journal pages 188-189. Have children share multiple solution strategies and write number models for each story. You may also wish to discuss how children knew they were on the right track in solving the problem, based on estimating or determining a reasonable answer in another way.” 
• MP6: In Lesson 2-7, Focus: Introducing Unit Boxes, students practice and use a unit box to identify the object being counted, “In Everyday Mathematics, a device called a unit box is often used to show the unit. Display a unit box and write the word ‘pennies’ in it. Tell children that they will count pennies and that the unit box is a reminder of what they will be counting.” “Have children use any of the appropriate units - pennies, cents, and ￠ - to label the counts.” 
• MP7: In Lesson 9-9, Focus: Making Up and Solving Number-Grid Puzzles, students use place value understanding to create and solve number-grid puzzles. “Have children fold Math Masters, page TA43 into four equal parts. Invite children to make up number-grid puzzles that their partners can solve using their understanding of tens and ones patterns on the number grid. Children draw around some of the grid cells on one part of the sheet to make a puzzle piece and then write a 2-digit number in one of the cells. They trade puzzles with their partners who then fill in all of the missing numbers.”
• MP8: In Lesson 6-6, Focus: Math Message and Developing the Making-10 Strategy, students use double ten frames to make teen numbers, “How could you show 15 on a double ten frame? How about 18? 19? Record your thinking on your slate. Have children use their own double ten frames and counters to share their thinking. Although children may represent the numbers in the Math Message in a variety of ways, emphasize filling one ten frame with ten and then representing the remaining ones on the second ten frame. Generalize this idea by asking: How can you represent teen numbers?” 

Examples of the materials not attending to the full meaning of MP5 because students do not get to choose tools strategically include:

• In Lesson 1-5, Focus: Telling 1-More and 1-Less Stories, students use a number line to solve number stories, “Have children use their own number lines as you solve other stories together. Or make up similar stories using the names of children in your class. Felipe walks 8 steps from the bathroom toward his bedroom. Begin with your finger on 0 on the number line. Then move your finger 8 steps or hops forward to model getting to the door of the bedroom. If he takes 1 more step, he will be inside his bedroom. Move your finger 1 more step to 9 to model getting inside the bedroom.” 
• In Lesson 4-3, Focus: Measuring with a pencil, students measure the lengths of objects by moving one unit, “Distribute one pencil to each child to use in measuring the tops of their desks. Have children explain how to make sure they get correct measurements. For example, ‘I put a finger where the pencil ends before I move it.’” 
• In Lesson 9-1: Focus: Measuring with Rulers, students discuss how to correctly use a ruler made of paperclips to measure, “Have children measure a few small classroom items. Demonstrate aligning objects at different starting points on the ruler. Discuss which alignments allow you to most easily count the number of paper-clip units to determine the length of the item.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Student materials consistently prompt students to construct viable arguments and analyze the arguments of others. Lessons offer Differentiated Options where students work in small groups or with partners as teachers facilitate discussions so students have opportunities to construct viable arguments and critique the reasoning of others. Open Response and Reengagement Lessons provide opportunities for students to critique the open response answers of other students. 

Students construct arguments. Examples include:

• In Lesson 1-8, Focus: Introducing Rock, Paper, Scissors, students play the game Rock, Paper, Scissors, and record the results in a tally chart, “Children play 20 rounds. After each round, the partnership makes a tally mark in the tally chart to indicate the winning gesture of a tie. Will one gesture win more often than the others? Why? Sample answer: No. If one gesture won more often, then players would always choose it and the game would always end in a tie.” 
• In Lesson 2-5, Focus: Find the Missing Day, students discuss strategies for determining whether any days of the week are missing from a journal page, “Are any days missing? Which days are missing? Tell your partner how you know.”
• In Lesson 4-4, Focus: Making Arguments, students share strategies and discuss the meaning of argument. Students respond to the Math Message in the Math Journal by circling which ribbon they believe is longer. “Have children share their answers for Problem 1. Ask volunteers to explain how they decided which ribbon was longer. Sample answer: The top one looked longer. I could fit more paper clips along the top ribbon. I used a string to show how long one ribbon was, then lined the string up with the other ribbon.”

Students critique the reasoning of others. Examples include:

• In Lesson 4-4, Measuring a Marker, Getting Ready for Day 2, “Show a paper with an argument that reveals a misconception. For example, a child might argue that Sofia is the best measurer because she used the most blocks, or argue that Li or Sofia is the best measurer because they lined up the blocks with both ends of the marker, as shown on Child A’s paper. Ask: What was this child’s thinking?”
• In Lesson 6-4, Focus: Developing the Near-Doubles Strategy, students use doubles facts to help them find other sums, “After the second image (Quick Look Card 81), ask children to share what they saw and how they saw it. Allow children to share multiple strategies, but highlight starting from the double 4 + 4 = 8 and adding 1 more dot to make 9. Emphasize this strategy by following up with questions such as Did everyone understand Damon’s strategy? and Can someone else explain it again for us? Prompt the class to apply it by asking Can you use Damon’s strategy on this next one?”
• In Lesson 7-10, Practice: Math Boxes, Item 5, students are shown a number box with the expression 3 + 4, and asked, “Raol wants to show (picture of 7 longs) in the box. Is that right or wrong? Explain. Sample answer: That’s wrong because that shows 7 tens and (picture of 7 cubes) is 7 ones.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Examples of assisting teachers in engaging students to construct viable arguments include:

• In Lesson 5-1, Focus: Introducing The Digit Game, students play a game to practice comparing numbers using place value, “Ask children to make arguments for why one number is larger than the other based on the number of tens or ones each digit represents. Observe: Which children made the larger number from their two digits? Which partnerships correctly identified the larger numbers? Discuss: Should you put your larger number card in the tens or ones place? Why? If your number has a larger digit in the ones place than your partner’s number, but your partner’s number has a larger digit in the tens place than your number, whose number is larger?”
• In Unit 6 Organizer, Mathematical Background: Process and Practice, Standard for Mathematical Process and Practice 3, “Unit 6 offers many opportunities for children to ‘construct viable arguments and critique the reasoning of others,’ as stated in Mathematical Process and Practice 3. As children respond to prompts such as How did you get that answer?, Explain your thinking, or Do you agree with what Maria just said? Why?, they are encouraged to demonstrate their thinking and make sense of others’ ideas. For your children, clear mathematical communication relies on concrete objects as much-if not more than-verbal explanations.”
• In Lesson 8-2, Focus: Partitioning Pancakes in Halves, students divide pancakes into two equal shares and discuss how they know the shares are equal, “Ask partners to discuss how they can show that their shares are equal, and have several partnerships share their ideas. Sample answer: I put one piece on top of the other, and they are the same size. Show examples from children who created unequal shares, and discuss whether these are equal shares of the pancake.”

Materials assist teachers in engaging students to analyze the arguments of others frequently throughout the program. Examples include:

• In Lesson 4-7, Focus: Introducing Addition Doubles, students are introduced to doubles, “Have partnerships further explore doubles by completing journal page 57. Before they begin, elicit and record an example for each column. Explain that they do not need to record sums. As children work, record several of their “Not Sure” expressions to discuss later. These may include larger addends (10 + 10 or 1,000 + 1,000) or addends that are closed or look similar (21 + 23 or 13 + 31). Have children share their conjectures as to whether these Not Sure expressions are actually doubles. This analysis will help them solidify their definitions of doubles.”
• In Lesson 5-12, Adding Animal Weights, Getting Ready for Day 2, students review each other’s work to the Open Response and discuss, “Show work where a child used a tool inefficiently or found an incorrect answer, as in Child A’s work. Child A tried to find the total weight of the first-grade girl (41 lb) and the squirrel (4 lb) by counting by 1s on the number grid and obtained an incorrect answer. Ask: How did this child use the number grid? What would you do the same or differently? Why?”
• In Lesson 6-8, Pencils for the Writing Club, Getting Ready for Day 2, students review each other’s work to the Open Response and discuss, “Show a paper on which a child wrote a number for the answer to Problem 1, as in Child A’s work. Have children talk to a partner about whether the answer makes sense for the question. Ask: What is Problem 1 asking you to do? What did this child do to find the answer to Problem 1? Does 15 seem like a good answer? What do you think the answer to Problem 1 should look like? A number? A word? Sample answer: 15 doesn’t seem right because the answer should be yes or no.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 1 partially meet expectations for explicitly attending to the specialized language of mathematics. The materials provide explicit instruction on how to communicate mathematical thinking using words, diagrams, and symbols, but there are instances when the materials use mathematical language that is not precise or appropriate for the grade level.

The Section Organizer provides a vocabulary list of words to be used throughout lesson discussions. Each lesson contains a vocabulary list, Terms to Use, and vocabulary words appear in bold print in the teacher notes. Some lessons incorporate an Academic Language Development component that provides extra support for the teacher and students. Additionally, the Teacher’s Lesson Guide contains a detailed glossary with definitions and images where appropriate. Examples of explicit instruction on how to communicate mathematical thinking include:

• In Lesson 1-8, Focus: Organizing and Representing Data in a Tally Chart, students select a data collection topic, create a tally chart, and answer questions about their data, “Remind children that they made tally charts to model how they sorted animals. Explain that today they will use a tally chart to represent real-world data, information they gather by counting, asking questions, or observing.”
• In Lesson 4-1, Academic Language Development, “Explain that when you describe how long an object is, you are talking about the length of an object. Provide children with sentence frames for both words, such as, ‘My book is about four crayons long. The length of my book is about four crayons.’”
• In Lesson 4-4, Focus: Making arguments, students share and discuss their work from the Open Response question, “Tell children that when they explained how they knew which ribbon was longer and which number was larger, they were making arguments. Explain that in mathematics, the word argument does not mean a disagreement or fight. Instead, a mathematical argument is a special kind of explanation mathematicians use to convince people they are right. Have children repeat a few of their strategies for the Math Message Problems. After each strategy ask: Does this convince you that the top ribbon is longer or that the first number is larger?”
• In Lesson 5-9, Academic Language Development, “Clarify the meanings of then and than. Display the words to show the spellings. Explaining that the word then has to do with time and answers the question ‘When?’ Then is spelled like when. The word than is a comparison word and is used with more than and less than, such as 75 is more than 56.”
• In Lesson 9-1, Focus: Making Rulers, students discuss how to use a ruler to measure and practice using it to measure objects, “Tell children that today they will make a special collection of units called a ruler. Children may be familiar with rulers and the terms inch and foot. Explain that a ruler can be made with any unit. In second grade, they will use inches and feet as units. But today they will make rulers using a unit they have used often in first grade-the paper clip.”

Examples of the materials using mathematical language that is not precise or appropriate for the grade level include:

• In the Student Reference Book, “The turn-around rule, says you can add two numbers in either order. Sometimes changing the order makes it easier to solve problems. Example: If you don’t know what 3 + 8 is, you can use the turn-around rule to help you, and solve 8 + 3 instead. 8 + 3 is easy to solve by counting on.” 
• In the Student Reference Book, “In a Frames-and-Arrows diagram, the frames are the shapes that hold the numbers, and the arrows show the path from one frame to the next. Each diagram has a rule box. The rule in the box tells how to get from one frame to the next.” 
• In the Student Reference Book, “A function machine uses a rule to change numbers. You put a number into the machine. The machine uses the rule to change the number. The changed number comes out of the machine.”