3rd-5th Grade - Gateway 2

The materials reviewed for Experience Math Grades 3 through Grade 5 meets expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

Conceptual understanding is developed throughout the course using a three-part lesson structure. Each lesson has a Mind’s On Activity, an Action Task, and a Consolidate Activity. During teacher guidance in the Action task, the "And the Point Is" notes clarify the specific conceptual goal behind each activity. Students have individual opportunities to demonstrate conceptual understanding through performance tasks, wonder tasks, and your turn problems. Conceptual understanding is assessed through math routines such as Math Congress and Gallery Walks, Exit Tickets, and Self/Peer Assessment. Examples include:

• Grade 3, Topic 4: Representing Numbers, Lesson 4, Minds On Activity, students develop conceptual understanding of fractions as numbers on the number line by reasoning about the location of fractions and explaining their placement. Question 1 states, “Which letter do you think shows \frac{3}{4} on this number line? Why?” Students are provided with a number line with three points labeled A, B, and C. Point A is at about \frac{1}{2}, Point B at about \frac{3}{4}, and Point C at about \frac{7}{8}. Teacher Guidance, After the Minds on Activity states, “Show students an open number line. Mark 0 and 1 on the number line. Ask students where they would mark the fraction 12  on this line. Ensure that students realize that \frac{1}{2} is in the middle of the line. You might want to color a rectangle from 0 to 12 to highlight the connection to a fraction of a length or area. Then show students a number line divided into three equal parts. Point to the tick marks on the number line, and ask students what fractions the tick marks represent. [13, 23]Draw students’ attention to the fact that \frac{2}{3} is actually two \frac{1}{3} sections, or 2 sets of \frac{1}{3}. The tick marks also tell what fraction of the number line from 0 to 1 ends at those points.” Action Task, Question 2 states, “Estimate where each fraction is on the number line.” Students are provided number lines labeled with endpoints 0 and 1. They plot the fractions \frac{2}{4}, \frac{5}{8}, and \frac{1}{6} on separate number lines. Consolidate Questions, Question 1 states, “Why does it make sense that the numbers between 0 and 1 on a number line are fractions?” Question 5 states, “Do you think the fraction \frac{2}{3} is a number? Explain.” (3.NF.2)

• Grade 4, Topic 12: Modeling Multiplication of Two-Digit Numbers, Lesson 2, Additional Practice, students develop conceptual understanding of multiplying two-digit numbers by estimating, modeling, and using the model to reason about the accuracy of their product. Question 1-3 states, “1. Estimate the product of 27\times33. 2. Create a model for 27\times33. Sketch your model. 3. Use your model to calculate 27\times33. Is the product close to your estimate?”(4.NBT.5)

• Grade 5, Topic 5: Fraction Operations, Lesson 1, Minds On Activity, students develop the conceptual understanding of using common denominators. Question 1 states, “Suppose you had 28 baskets of 6 apples and 5 baskets of 9 apples. How could you quickly find how many baskets of 3 apples you could create?” Teacher Guidance, In This Activity … states, “Students consider how many baskets of three apples they can make from a number of baskets of six apples and baskets of nine apples. They use renaming to help them solve the problem. Teacher Guidance, And the Point Is … states, “This Minds On Activity helps students realize that thinking in a different unit can simplify the problem and leads nicely into the idea of renaming to simplify calculations.” Action Task, Questions 3-4 states, “For 3 and 4, what is a reasonable estimate for the sum? Explain your reasons. 3. \frac{3}{4}+\frac{4}{5} 4. \frac{1}{3}+\frac{3}{5}” (5.NF.1)

The materials reviewed for Experience Math Grades 3 through 5 meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

The materials include multiple pathways for fluency development, such as Number Talks and Data Talks, which target fluency and flexibility in thinking about numbers and operations while allowing students to use their own problem-solving strategies. Students engage in practice through Your Turn activities that provide procedural practice opportunities. Fluency is embedded within the three-part lesson structure, Minds On, Action Task, and Consolidate, and is assessed within the program’s assessment system, which includes specific skill and concept questions that help teachers monitor and support students’ procedural fluency development. Examples include:

• Grade 3, Topic 3: Multiplying and Dividing Whole Numbers within 100, Lesson 4, Additional Practice, Questions 4-7, students demonstrate procedural skill and fluency with multiplying and dividing within 100. The directions state, “Find the product of each multiplication fact. 4. 6\times7 5. 7\times9 6. 8\times4 7. 9\times6” (3.OA.7)

• Grade 4, Topic 20: Adding and Subtracting Whole Numbers, Lesson 3, Your Turn, Questions 4-7, students demonstrate procedural skill and fluency in adding and subtracting whole numbers using standard algorithms. The directions state, “Calculate each sum using the standard algorithm. Estimate to check. 4. 3,148 + 5,893 5. 4,093 + 2,258 6. 658+3,772 7. 14,256 + 7,386” (4.NBT.4)

• Grade 5, Topic 7: Whole Number Operations, Lesson 1, Action Task, students develop procedural skills and fluency as they multiply multi-digit whole numbers using the standard algorithm. The directions state, “1. You multiply a three-digit number by 6. Choose a value for the three-digit number so that the product is between 1,500 and 3,000. Your three-digit number must use some of the digits shown above. Use the standard algorithm to show that your choice makes sense. 2. You multiply a two-digit number by a two-digit number. Choose a value for each two-digit number so that the product is between 4,000 and 5,000. Choose the digits from the choices above. Use the standard algorithm to show that your choice makes sense.” Teacher Guidance, Conversation Starters states, “Could the three-digit factor be in the 700s? How do you know? Are there any missing digits you can be sure of right away? Why? Why do you think a 3 goes there? How did you make sure your example had all the digits missing?” (5.NBT.5)

The materials for Experience Math Grades 3 through 5 meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

Multiple routine and non-routine applications of mathematics are included throughout the grade level, with single- and multi-step application problems embedded within the three-part lesson structure, Minds On, Action Task, and Consolidate. Students engage in Wonder Tasks, Making Connections Tasks, and Brain Benders. Wonder Tasks are three-act, problem-based activities in which students formulate questions about situations and communicate their thinking. Making Connections Tasks integrate concepts from multiple domains and provide opportunities for students to apply mathematical skills and concepts while solving complex, multi-part problems. Brain Benders present problem-solving opportunities in real-world contexts. Examples include:

• Grade 3, Topic 3: Multiplying and Dividing Whole Numbers within 100, Lesson 5, Your Turn, Your Turn Questions, students apply their understanding as they solve multistep word problems involving equal groups to determine and explain differences in quantities and costs. Question 1 states, “Klaudia bought 3 packages of apples. Each package has 6 apples and costs $4. Olivia bought 5 packages of the same apples. a. How many more apples did Olivia buy than Klaudia? Explain. b. How much more money did Olivia spend on the apples than Klaudia? Explain.” (3.OA.8)

• Grade 4, Topic 8: Fraction Operations, Lesson 4, Action Task, students apply their understanding of fractions as they use money to represent tenths and hundredths of a dollar and explain their reasoning. The Action Task states, “Morgan has some dimes and fewer than 10 pennies. Simone has some dimes and fewer than 10 pennies, too. 1. Complete each part below. a. Together Morgan and Simone have 85\text{¢}. How many dimes and how many pennies might each have? b. How could you write Morgan’s amount of money as \frac{\square}{10} of a dollar \frac{\square}{100} of a dollar?” Teacher Guidance, And the Point Is… states, ”This Action Task uses money as the context, since it’s one of the places that tenths and hundredths realistically come up; leads nicely into work using decimal notation for money; lets students choose the numbers of pennies and dimes, which will allow a richer Consolidate discussion to help students draw conclusions from the task.” (4.NF.3.d)

• Grade 5, Topic 4: Whole Number and Fraction Operations, Lesson 4, Action Task, students apply their understanding of fractions as they solve real-world problems involving division of whole numbers by unit fractions. Question 1 states, “Damian has 8 cups of rice. His recipe calls for \frac{1}{4} of a cup of rice. How many times can he make the recipe before he runs out of rice?” (5.NF.7.c)

The materials reviewed for Experience Math Grades 3 through 5 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 as reflected by the standards.

Multiple aspects of rigor are engaged simultaneously across the materials to develop students' mathematical understanding of individual lessons. Each topic within the curriculum supports a variety of instructional approaches that incorporate conceptual understanding, procedural fluency, and application in a balanced way. 

For example:

• Grade 3, Topic 5: Fraction Equivalence and Comparison, Lesson 2, Minds On Activity, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they compare fractions with the same denominator or the same numerator and apply their understanding to authentic contexts. Question 1 states, “Fraction strips are placed in three rows. The row showing \frac{4}{6} is longer than the rows showing these fractions: \frac{2}{} \frac{3}{} What could the denominators be? How do you know 46 is more than each of these fractions?” Action Task, Part 1, Directions states, “For Questions 1 to 3, use fractions to compare how many of the shelves in the bookcases have books on them. Write an inequality that compares the fractions.” For Questions 1–3, students are provided with two bookcases that are filled differently. Teacher Guidance, Conversation Starters states, “What information about the bookcase tells you the denominator? The numerator?” Consolidate Questions, Question 1 states, “How did you decide what fraction to use to represent how many shelves in each bookcase had books on them?” Question 3 states, “How does representing the fractions on a number line help you see which one is greater?” Question 6 states, “How could you explain to someone why \frac{2}{6} is less than \frac{2}{3} but more than \frac{2}{8}?” (3.NF.3d)

• Grade 4, Topic 19: Estimating and Comparing Five-Digit and Six-Digit Whole Numbers, Lesson 3, Action Task: students demonstrate conceptual understanding and procedural fluency as they order large numbers. Students are provided with a list of basketball arenas and football stadiums with their seating capacities. They use this list to answer questions. Action Task states, “1. Put the seating capacities in order from least to greatest. Explain your thinking. You can use place-value charts or number lines to help you. 2. Create a set of five or six numbers to order. a. Follow this criteria to make the numbers. Make two numbers with six digits and three numbers with five digits. Each number must use only the digits 4, 0, and 5. b. Put your numbers in order from least to greatest. At least two of the numbers must start with the same two digits.” (4.NBT.2)

• Grade 5, Topic 5, Fraction Operations, Lesson 4, Action Task, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they add and subtract fractions and mixed numbers. Question 1 states, “You have 4\frac{1}{3} quarts of pineapple juice. You use 1\frac{1}{2} quarts. Are there more or less than 3 quarts left? How much more or less? Write equations to show how you solved the problem. Estimate to show that your answer makes sense.” (5.NF.2)

The materials reviewed for Experience Math Grades 3 through 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students across the 3-5 grade band engage with MP1 throughout the year. MP1 is found in the three-part lesson structure as students make sense of problems during Minds On discussions, persevere through open-ended Action Tasks, and reflect on strategies in the Consolidate portion of lessons. Essential Understandings connect new learning to big ideas, helping students analyze problem information and plan solution pathways, while Parallel Tasks provide differentiated entry points that allow all learners to engage in productive struggle. Students regularly interpret conditions in real-world problems, compare quantities, and choose tools to solve problems, while teacher supports such as prompts, embedded coaching videos, and “Probing and Extending” questions guide students in evaluating their solutions and asking “Does this make sense?” 

Across the grades, students engage in open-ended tasks that support key components of MP1, including sense-making, strategy development, and perseverance. These tasks prompt them to make sense of mathematical situations, develop and revise strategies, and persist through challenges. They are encouraged to analyze problems by engaging with the information and questions presented, using strategies that make sense to them, monitoring and evaluating their progress, determining whether their answers are reasonable, reflecting on and revising their approaches, and increasingly devising strategies independently. 

An example in Grade 3 includes:

• Topic 7: Length and Time, Lesson 1, Student Experience Book, students interpret the context of elapsed time, represent the starting and ending times on both analog and digital clocks, and determine a new time. They explain their thinking by answering questions related to the time. Students evaluate whether their answers make sense by comparing their analog and digital representations and justifying their reasoning. Teacher Experience Guide, Action states, “Students make sense of and solve problems about how much time has passed.” Action Task states, “4. What time is it if it’s 42 minutes after 2:43? Show both times on both an analog clock and digital clock. Explain how you figured out the time.” Teacher Experience Guide, Using the Action Task states, “Because of the flexibility of the task, some students might choose times within the same hour, but others might not. Although normally students are expected to read time and not set time, there is value in providing the opportunity to choose the times. While only limited accuracy is required, students should be encouraged to make the minute hand longer than the hour hand and place the hour hand in a reasonable position given the time selected.” Teacher Guidance, Connecting Ideas and Experiences states, “Ask students what time they leave their house for school in the morning. Then ask how they know that they need to leave at that time. If students take a bus, ask how long it takes to walk to their bus stop. If they get a ride from an adult, ask about how long the drive is to school. If they walk, ask about how long the walk is.” Teacher Guidance, Conversation Starters states, “When are the hands of a clock close together? How might you use a number line to help you figure out how much time has passed? Is 42 minutes more or less than an hour? When might the hands be almost opposite each other?” 

An example in Grade 4 includes:

• Topic 12: Multiplying Two-Digit Numbers, Lesson 4, Student Experience Book, students solve a word problem involving decimal numbers, making sense of the problem and finding different possible solutions to the situation. Teacher Experience Guide, Action states, “Students must recognize that the situation calls for multiplication and possible subtraction as well, but maybe not.” Teacher Experience Guide, Action Task, Conversation Starters states, “Why would you think about multiplying to solve the reading books problem? Would you multiply the 124 by 2 or 3? Why? When you are creating your own problem that involves multiplication, what might it be about?” Student Experience Book, Action Task, “1. The students in Aidan’s class read 124 books in one month. The class goal is to read a total of 3 times that many books by the end of three months. a. How many more books will they have to read over the next two months? Explain your thinking. b. Explain how you know this problem is a multiplication situation. 2. Choose three pairs of numbers. Create and solve a multiplication problem for each pair of numbers you chose. For one or two of your problems, include one more number and an extra computation that might be addition, subtraction, division, or another multiplication. Describe or show your calculation strategies. Estimate to check your answers. Show how you estimated. Explain why multiplication (and any other operation, when you involve other operations) can be used to solve your problem. Create a different problem for the same pair of numbers.” Students see the following pairs of numbers: 4 and 152; 10 and 30; 12 and 24; 25 and 80; and 36 and 19.

An example in Grade 5 includes:

• Topic 12: Adding and Subtracting Decimals, Lesson 2, Student Experience Book, students solve a word problem involving decimal numbers, making sense of the problem and finding different possible solutions to the situation. Teacher Experience Guide, Action states, “Students must persevere to consider alternative solutions to the given problem.” Teacher Guidance, Action, and the Point is… states, “It is important for students to realize that there are many solutions. There is also room for interpretation about what a ‘few’ pans of lasagna means. Allow for lots of latitude in this interpretation.” Student Experience Book, Action Task states, “Families made 9 pans of lasagna for a school event. Each pan of lasagna was cut into 10 equal pieces. After the event there were a few full pans of lasagna left over, 3 pieces of another pan of lasagna left over. 1. How much lasagna might have been eaten? List lots of different possible answers. Write your answers in the following form using decimals: \square.\square pans of lasagna. Show or explain how you figured out your answers. 2. What equation might you have written to describe one of your problems?”

The materials reviewed for Experience Math Grades 3 through 5 meet expectations for supporting the intentional development of MP2: Reason abstractly and quantitatively, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 3-5 grade band engage with MP2 throughout the year. MP2 is found in the three-part lesson structure as students move between concrete, pictorial, and abstract representations during Minds On discussions, reason about quantities and relationships in open-ended Action Tasks, and justify strategies in the Consolidate portion of lessons. Essential Understandings connect new learning to big ideas, supporting students in decontextualizing and contextualizing problems, while Parallel Tasks provide differentiated entry points that allow all learners to represent quantities flexibly. Students regularly create and interpret representations, consider units, and explain the meaning of quantities, while teacher supports such as prompts, embedded coaching videos, and “Probing and Extending” questions guide students in deepening their reasoning and making connections between mathematical ideas 

Across the grades, students participate in tasks that support key components of MP2, including reasoning with quantities, representing situations symbolically, and interpreting the meaning of numbers and symbols in context. These tasks encourage students to consider the units involved in a problem, analyze the relationships between quantities, and connect real-world scenarios to mathematical representations. Teachers are guided to support this development by modeling the use of mathematical notation, asking clarifying and probing questions, and facilitating conversations that help students make connections between multiple representations. 

An example in Grade 3 includes:

• Topic 17: Liquid Volume, Lesson 2, Student Experience Book, students make predictions about how much liquid common items can hold and then verify their predictions using known facts. Student Experience Book, Action Task states, “Two dragons are camping. They will collect water from the nearby river. The big dragon needs 5 liters of water. The small dragon needs 2 liters of water. 1. Choose one dragon to collect water for. Predict which containers you could use to collect about the right amount of water for the dragon you chose. You are allowed to use a container more than once. Do not use the liter container until you are testing your prediction in Question 2.” Teacher Experience Guide, Action states, “Students use quantitative reasoning to make estimates about which bowls hold a certain amount of liquid.” Teacher Experience Guide, And the Point Is… states, “This Action Task is engaging and interesting for students by providing a fun context about dragons and camping, which also provides a purpose for measuring. Encourages students to look at relationships between various measurements by using different container sizes. It will be easier for students if they start over in the event when they add too much water because it is likely they will lose track of how many containers they have added if they start removing containers full of water.” Teacher Experience Guide, Connecting Ideas and Experiences states, “To encourage students to share their own unique perspectives, you can do the following: Ask students if they have ever gone camping. Ask the students who have gone camping what it was like. Ask the students who have not gone camping if they’ve seen a movie or TV show where people went camping. Discuss containers that might be available at a campsite (e.g., coolers, buckets, jugs, pitchers, garbage bins). Classify the containers you name as being able to hold a small liquid volume or a large liquid volume.”

An example in Grade 4 includes:

• Topic 5: Multiplying by One-Digit Numbers, Lesson 1, Student Experience Book, students decide when multiplication could be used to solve word problems and reason abstractly if their strategy makes sense. Student Experience Book, Action Task states, “Sakura bought a game for $3 and a toy for $4 at a yard sale. How much did she spend? Pedro bought a toy that costs 3 times as much as the $4 toy he already had. How much did the new toy cost? 1. Which problem from above could you solve by multiplying? How do you know? a. Write a multiplication equation for it and solve it. b. Rewrite the other problem so that it is a multiplication problem. Then write a multiplication equation for it and solve it.” Teacher Experience Guide, Action states, “Reason Abstractly, Students relate contextual situations to appropriate mathematical situations.” Teacher Experience Guide, In This Task… states, “Students decide if given problems can be solved by multiplying, and they rewrite problems that are not multiplication problems so that they can be solved by multiplying.” 

An example in Grade 5 includes:

• Topic 2: Representing Multiplication and Division of Fractions, Lesson 1, Student Experience Book, students add fractions with unlike denominators using models and equivalent fractions. They also estimate sums of fractions and use reasoning to determine the types of situations in which they would add fractions. Student Experience Book, Action Task states, “Some students each started reading a book on Monday. This chart shows the fraction of the book read each day so far.” Students see a table with each student’s name and the fraction of the book read on Monday and Tuesday. “1. Choose data for at least two of the students. For each student, use benchmark fractions to estimate the fraction of the book read on both days together. Explain your estimate. 2. Figure out an exact answer to the problem in Question 1 using fraction strips. Write an addition equation for each student that you chose.” The Teacher Experience Guide states, “Students use mathematical symbols to estimate sums and then apply that thinking to the context of the problem.”

The materials reviewed for Experience Math Grades 3 through 5 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 3-5 grade band engage with MP3 throughout the year. MP3 is found in the three-part lesson structure, students share and justify their reasoning during Minds On discussions, construct arguments and solve open-ended problems in Action Tasks, and critique the reasoning of others in the Consolidate portion of lessons. Open questions provide multiple entry points and create opportunities for students to explain their thinking and respond to peers, supporting the development of mathematical communication skills. Structured routines such as Math Congress and Gallery Walks provide spaces for students to examine, compare, and critique mathematical strategies. Students regularly explain why their reasoning is mathematically valid and analyze the reasoning of others, while teacher supports such as prompts, embedded coaching videos, and “Probing and Extending” questions guide students in deepening their arguments and making connections between mathematical ideas.

Across the grades, students participate in tasks that support key components of MP3. These include constructing mathematical arguments, analyzing errors in sample student work, and explaining or justifying their thinking orally or in writing using concrete models, drawings, numbers, or actions. Students are encouraged to listen to or read the arguments of others, evaluate their reasoning, and ask clarifying questions to strengthen or improve the argument. They also have opportunities to make and test conjectures as they solve problems. Teachers are guided to support this development by providing opportunities for students to engage in mathematical discourse, set clear expectations for explanation and justification, and compare different strategies or solutions. Teachers are prompted to ask clarifying and probing questions, support students in presenting their solutions as arguments, and facilitate discussions that help students reflect on and refine their reasoning.

An example in Grade 3 includes:

• Topic 9: Estimating Whole Numbers, Lesson 1, Student Experience Book, students represent numbers that satisfy specific place-value conditions by using base-ten blocks and number lines. Student Experience Book, Action Task states, “Zuri and Jordan are competing on a quiz show. They each have 5 questions to complete. Whoever answers all questions correctly and has the greatest difference between the greatest and least numbers in their answers wins the competition. 1. Who do you think will win? Explain.” Teacher Experience Guide, Action states, “Students construct arguments to defend their predictions on who will win.” Teacher Experience Guide, Using the Action Task states, “Students might benefit from modeling the numbers with base-ten blocks, number lines, and/or place-value charts and counters.” Teacher Experience Guide, Connecting Ideas and Experiences states, “To encourage students to share their own unique perspectives, you can do the following: Ask students about the games they like to play with family or friends or the game shows they like to watch on TV. Ask whether any of the games they like to play or watch have players solve math questions.” Teacher Experience Guide, Conversation Starters states, “Could the digit in the hundreds place be 9? What are some possibilities for the digit in the tens place? Why those? If the digit in the ones place is the greatest, what do you know about the digit in the hundreds place?” Teacher Experience Guide, Grouping states, “Pairs can share materials, make comparisons together, and complete one worksheet.” Teacher Experience Guide, Success Criteria states, “Share or co-construct Success Criteria 1 to 3 during the Action Task. Withhold Criteria 4 to 6 until after discussing the Consolidate Questions.”

An example in Grade 4 includes:

• Topic 17: Representing Five-Digit and Six-Digit Whole Numbers, Lesson 4, Student Experience Book, students use information about a given number to figure out other information about that mystery number. Students provide reasons that serve as viable arguments for what the mystery number is. Student Experience Book, Action Task states, “Here are some clues about a mystery number. It’s between 5,000 and 10,000. It has a 7 in the hundreds place or the thousands place. It can be written as 🗇 thousands + 🗇 tens. 1. Name two or more numbers that the mystery number could be. Explain how you know each number matches the clues. 2. Name two or more numbers that the mystery number could not be. For each number, explain how you know that it does not match the set of clues.” Teacher Experience Guide, Action states, “Students use this practice when they decide what numbers are possible or impossible for a given set of clues.” Teacher Experience Guide, And the Point is… states, “In this Action Task students are given clues about a number. Often when students do this kind of task, they focus on guessing the number. For Question 1, discourage that at first, and instead focus the students on thinking about what else they are sure about. Only then should they think about numbers that are possible.” 

An example in Grade 5 includes:

• Topic 5: Fraction Operations, Lesson 5, Student Experience Book, students explore three conjectures related to changing numerators and denominators of fractions being multiplied. Student Experience Book, Action Task states, “Do you agree or disagree? Why? 1. You start with \frac{a}{b}\times\frac{c}{d}. Conjecture: If you reduce a by 1 and b by 1, the product always decreases. 2. You start with \frac{a}{b}\times\frac{c}{d}. Conjecture: If you reduce a by 1 and increase c by 1, the product could sometimes decrease, but not always.” Teacher Experience Guide, Action states, “Students construct arguments for agreeing or disagreeing with three conjectures related to multiplying fractions.” Teacher Experience Guide, Conversation Starters states, “What different sizes of fractions do you think you might try? Could you have predicted whether that product would likely increase or decrease or not? Do you think you should also try fractions closer to 0?”

The materials reviewed for Experience Math Grades 3 through 5 meet expectations for supporting the intentional development of MP4: Model with mathematics, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 3-5 grade band engage with MP4 throughout the year. MP4 is found in the three-part lesson structure as students identify real-world problems and create mathematical models during Minds On discussions, apply and test models in Action Tasks, and refine their models in the Consolidate portion of lessons. Wonder Tasks, Making Connections Tasks, and Brain Benders provide authentic contexts where students select and use mathematical tools and representations to solve problems. Students regularly build and interpret models with physical and virtual manipulatives, while teacher supports such as embedded coaching videos, “And the Point Is” sections, and “Probing and Extending” questions guide students in making connections between models and mathematical ideas.

Across the grades, students participate in tasks that support the intentional development of MP4, including putting problems or situations in their own words and identifying important information, using the math they know to solve problems and everyday situations, modeling the situation with appropriate representations and strategies, describing how their model relates to the problem situation, and checking whether their answer makes sense, revising the model when necessary.

An example in Grade 3 includes:

• Topic 3: Multiplying and Dividing Whole Numbers within 100, Lesson 3, Student Experience Book, students identify ways to decompose a dividend into parts to solve a division problem. Student Experience Book, Action Task states, “The librarian has these new books. He asked Oliver to make packages with the same number of books in each package. 1. Oliver put the information books and fairy-tale books into 5 equal packages. Each package had the same number of each kind of book. a. Use counters to represent the problem. b. Write a division equation that tells how many books were in each package. c. How many of each type of book were in each package?” Teacher Experience Guide, Action states, “Students model a real-life situation with a mathematical equation.” Teacher Experience Guide, Using the Action Task states, “This Action Task provides several opportunities for students to use the Distributive Property with division. Students write division equations to solidify the links between sharing and division and represent the situations with counters to help them see why breaking up the dividend makes sense.” Teacher Experience Guide, Connecting Ideas and Experiences states, “Ask students what kinds of books they like to read. Students might respond with genres or series of books. Have students discuss the types of books they borrow from the library. Have students ever borrowed a nonfiction book? Discuss the different genres that students volunteer. You could create a chart to show favorites.” Teacher Experience Guide, Conversation Starters states, “Why is this a division situation? How might using an array of counters be helpful? What will you divide by? Why? Would 15 be a good number of books to use? Why or why not?”

An example in Grade 4 includes:

• Topic 20: Adding and Subtracting Whole Numbers, Lesson 1, Student Experience Book, students use mental strategies to help add and subtract two-digit and three-digit numbers. Student Experience Book, Action Task states, “You’re shopping for winter clothes. Here are some items to choose from. Use mental math for each calculation. Describe your strategies in words or pictures. 1. Choose two of the items to buy. The total cost must be less than 100. a. How do you know you chose two items that meet that requirement? b. Calculate the total cost of the two items. c. Calculate the difference between the two prices. d. Calculate the change you would get from a 100 bill for one of the items. 2. Choose two more of the items to buy. One can be an item you used in Question 1. a. Calculate the total cost of the two items. b. Calculate the difference between the two prices. c. Calculate the change you would get from a 100 bill for one of the items.” Students see a list of items and their prices: ice skates, $29; gloves, $11; coat, $69; boots, $37; and snow pants, $57. Teacher Experience Guide, Action states, “Students use mathematical representations to solve a real-world problem.” Teacher Experience Guide, Conversation Starters state, “What two items might cost about $50 together? When you figure out change, what operation do you use? Why?”

An example in Grade 5 includes:

• Topic 10: Representing Decimals, Lesson 3, Student Experience Book, students relate decimals and fractions to one another, converting fractions to decimals to the thousandths place or converting decimals to fractions with a denominator of 1,000. They also model the math and apply it to real-world situations. Student Experience Book, Action Task states, “Charlotte’s school has exactly 1,000 students. Groups of students can be described using decimals or fractions. 1. Choose two of these decimals: 0.115, 0.160, 0.425. 2. Describe what each decimal might represent about the students at Charlotte’s school. Make sure your choices make sense. 3. Use a model to show each decimal. Then write the decimal as a fraction.” Teacher Experience Guide, Action states, “Students model everyday situations using decimals.” Teacher Experience Guide, In This Task… states, “Students use a variety of models to relate decimals to thousandths and fractions, and they consider how those decimals and fractions might arise in a real-life situation.” Teacher Experience Guide, Using the Action Task states, “Provide thousandths grids and colored pencils, a base-ten large cube, and an assortment of flats, rods, and small cubes. You might review how to show one-tenth of the unit with each model (e.g., shade one column or ten hundredths in a thousandths grid, or show that one base-ten flat is one-tenth of a large base-ten cube). Have students also show one hundredth with each model. Then introduce the task for students to complete in pairs. Explain that they can decide what category of students each decimal or fraction might describe that makes sense. For example, they might decide that 0.115 describes the portion of students who are left-handed or that \frac{2}{5} describes what fraction of students has a certain pet or likes a certain food. You might want to review various ways to partition squares to show halves, fourths, fifths, tenths, and so on.”

The materials reviewed for Experience Math Grades 3 through 5 meet expectations for supporting the intentional development of MP5: Use appropriate tools strategically, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 3-5 grade band engage with MP5 throughout the year. MP5 is found in the three-part lesson structure as students access a range of physical and digital mathematical tools and learn to select and use them strategically based on context. Classroom manipulative kits from hand2mind are aligned with Brainingcamp virtual manipulatives, allowing students to move between concrete and digital representations. During Action Tasks, students choose tools such as base ten blocks, fraction tiles, or geometric instruments to support problem solving and reflect on their effectiveness. Embedded teacher supports, including Dr. Marian Small’s coaching videos, the Teacher Assistance Panel, “And the Point Is” sections, and “Probing and Extending” questions, guide teachers in helping students evaluate when tools are helpful and how they connect to mathematical ideas.

Across the grades, students participate in tasks that support key components of MP5. These include selecting and using appropriate tools and strategies to explore mathematical ideas, solve problems, and communicate their thinking. Students are encouraged to consider the advantages and limitations of various tools such as manipulatives, drawings, measuring devices, and digital technologies, and to choose tools that best support their understanding and reasoning. They have opportunities to use tools flexibly for investigation, calculation, representation, and sense-making. Teachers are guided to support this development by making a variety of tools available, modeling their effective use, and encouraging students to make strategic decisions about when and how to use tools. Teacher materials prompt opportunities for student choice in tool selection, promote discussion around tool effectiveness, and support the comparison of multiple tools or representations. Teachers are also encouraged to highlight how different tools can yield different insights and to help students reflect on their tool choices as part of the problem-solving process.

An example in Grade 3 includes:

• Topic 3: Multiplying and Dividing Whole Numbers within 100, Lesson 2, Student Experience Book, students use models, pictures, and patterns to explain multiplication by 0 and 1. Student Experience Book, Action Task states, “1. a. Continue this pattern. 5 × 2 = 10, 4 × 2 = 8, 3 × 2 = \rule{0.5cm}{0.15mm}, 2 × 2 = \rule{0.5cm}{0.15mm}, 1× 2 = \rule{0.5cm}{0.15mm} , 0 × 2 = \rule{0.5cm}{0.15mm}. b. How does the pattern work? c. Draw a picture to show what 0 × 2 means.” Student Experience Book, Action Task also states, “2. Look at the list of 5s facts. a. Continue this pattern. What would the next fact be? 5 × 5 = 25, 5 × 4 = 20, 5 × 3 = 15, 5 × 2 = 10, 5 × 1 = 5. b. How does the pattern work? c. Draw a picture to show what 5 × 0 means. 3. Use a model, pictures, or a pattern. a. Show what 7 × 1 means. Explain you're thinking. b. Show what 1 × 7 means. Explain you're thinking.” Teacher Experience Guide, Action states, “Students choose whether to use pictures, counters, or patterns to help them explain the idea of multiplying by 0 and 1.” Teacher Experience Guide, In This Task states, “Students explore the rules for multiplying by 0 and 1 using two different perspectives: models and patterns.” Teacher Experience Guide, And the Point Is… states, “This Action Task helps students understand that 0 × any number = 0, since no groups means there is nothing and any number 0 = 0 since groups of nothing means there is nothing. It helps students understand that 1 × any number = that number, since there is only one group and any number × 1 = that number since there are that many groups of 1.” Teacher Experience Guide, Conversation Starters states, “What makes this a pattern? How does that show 5 × 0? How does that show 1× 7?”

An example in Grade 4 includes:

• Topic 16: Weight and Mass, Lesson 1, Student Experience Book, students estimate and then measure the weights of several everyday items, and rename the weights of some items described in larger units as numbers of smaller units. Student Experience Book, Action Task states, “1. Estimate the weight of each object. Check using a pan balance: A plastic stapler, a book, a shoe, an eraser, and an empty ceramic mug. 2. Describe the weight of each object using pounds instead of tons. 3. Describe the weight of each object using ounces instead of pounds.” Teacher Experience Guide, Action states, “Students use the weights in efficient ways to determine weights of items.” Teacher Experience Guide, Using the Action Task states, “Ideally, students should work with pan balances and weight measures so they can really see what must be put together to balance various items. If this is not possible, small kitchen scales could be used. The disadvantage of the kitchen scale is that the numbers don’t really mean a lot to students; they don’t see where they came from.” Teacher Experience Guide, And the Point Is… states, “This Action Task is designed to provide hands-on experience for students in using ounces and pounds and help students develop a feel for how the number of units we use changes when moving from tons to pounds and from pounds to ounces.” Teacher Experience Guide, Conversation Starters states, “Do you think the book will weigh more than one pound or less than one pound? Which item do you think will be the lightest? The heaviest? Why might you need to use a few of the weights together on one side of the pan balance?”

An example in Grade 5 includes:

• Topic 10: Representing Decimals, Lesson 2, Student Experience Book, students choose a decimal to thousandths, represent it in a variety of ways, and describe what each representation shows about the number. Student Experience Book, Action Task states, “1. Choose one of these decimals to represent: 2.500, 0.750, 0.625, or 1.234. Represent the decimal in as many ways as you can. If you use a number line, you can choose your own start and stop numbers and labels. If you use base-ten blocks, you can allow any block to be 1 whole, and then you can name the other blocks accordingly. 2. Describe one thing that each representation clearly shows about your number.” Teacher Experience Guide, Action states, “Students use a variety of tools but focus on which ideas each tool best brings out about decimals.” Teacher Experience Guide, And the Point Is… states, “This Action Task allows students to choose numbers greater than or less than 1, numbers that can be described as known simple fractions (in this case \frac{3}{4} or \frac{5}{8}) or not, and numbers that have zeros in the thousandths place and numbers that don’t. It allows students to relate their representations to place-value ideas, as well as to known fraction or decimal values.” Teacher Experience Guide, Using the Action Task states, “Have blank number lines, thousandths grids, place-value charts, base-ten blocks, and counters for representing decimals available. Students can sketch their own open number lines or use the reproducible page Number Lines. If no students use base-ten blocks, you might model how to do that.” Teacher Experience Guide, Conversation Starters states, “How do you know that 2.500 is greater than 2? What fraction is the same or nearby? What decimals, that are more familiar, is it halfway between?”

The materials reviewed for Experience Math Grades 3 through 5 meet expectations for supporting the intentional development of MP6: Attend to precision, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 3-5 grade band engage with MP6 throughout the year. MP6 is found in the three-part lesson structure as students develop precision in communicating their mathematical thinking and justifying their solution approaches. Action Tasks provide opportunities for students to explain their reasoning and use mathematical language accurately. The Student Experience Book includes journal prompts that support reflection and articulation of ideas with clarity. During the Consolidate phase, students share strategies and refine explanations, while the teacher supports. Dr. Marian Small’s coaching videos, the Teacher Assistance Panel, “And the Point Is” sections, and “Probing and Extending” questions guide them to use precise vocabulary, attend to details, and connect strategies to mathematical ideas.

Across the grades, students engage in tasks that support key components of MP6. These include formulating clear explanations, using grade-level appropriate vocabulary and conventions, applying definitions and symbols accurately, calculating efficiently, and specifying units of measure. Students are also expected to label tables, graphs, and other representations appropriately, and to use precise language and notation when presenting mathematical ideas. Teachers support this development by modeling accurate mathematical language, ensuring students understand and apply precise definitions, and providing feedback on usage. Sessions include opportunities for students to clarify the meaning of symbols, evaluate the accuracy of their own and others’ work, and refine their communication. Teachers are encouraged to facilitate discussions and structure tasks that promote attention to detail in both reasoning and representation.

An example in Grade 3 includes:

• Topic 1: Measuring Area with Non-Standard Units, Lesson 3, Student Experience Book, students use a grid to compare the relative areas of the western states. Student Experience Book, Action Task states, “For Questions 1–6, use greater than ( > ), less than ( < ), or the equal sign ( = ) to compare the areas of the states. 1. area of Washington ___ area of Utah 2. area of California ___ area of Arizona 3. area of Nevada ___ area of New Mexico 4. area of Wyoming ___ area of Montana 5. area of Colorado ___ area of Nevada 6. area of Idaho ___ area of Utah”. Teacher Experience Guide, Action states, “Students accurately determine and compare the areas.” Teacher Experience Guide, Using the Action Task states, “Show students a map of the continental United States, focusing on the states in the West. Point out where various states are, such as Washington, Colorado, Wyoming, and California. Ask them which area looks biggest to them and which area looks smallest. Then show students the grid in the Action Task. Explain that the grid is what a map of the western states might look like if the entire continental Western United States were a square and the states were shown with smaller squares that represent their areas.” Teacher Experience Guide, Connecting Ideas and Experiences states, “Ask students about the size of the state they live in. Have them estimate and compare the size of their state to one of the states listed. If students live in one of the states listed, have them compare the size of their state to others given.” Teacher Experience Guide, Conversation Starters states, “How do you know the states are not all the same size? Which states look the biggest? The smallest? Which look close in size?” 

An example in Grade 4 includes:

• Topic 21: Length and Area, Lesson 3, Student Experience Book, students write sentences that include facts about lengths in meters, kilometers, feet, and miles, and then rewrite them using corresponding units. Student Experience Book, Action Task states, “1. Write a few sentences that include several facts about things that have lengths in meters. 2. Rewrite the sentences from Question 1 using only measurements in centimeters. 3. Write a few sentences that include several facts about distances in kilometers. The distances should be less than 100 kilometers. 4. Rewrite the sentences from Question 3 using only measurements in meters. 5. Write a few sentences that include several facts about distances in feet. Include some measurements that involve fractions. 6. Rewrite the sentences from Question 5 using only measurements in inches. 7. Write a few sentences that include several facts about distances in miles. Use some distances that are less than 25 miles. 8. Rewrite the sentences using only measurements in feet.” Teacher Experience Guide, Action states, “Students must be careful to associate the correct measures with the correct units.” Teacher Experience Guide, Using the Action Task states, “Internet access and/or books or fact sheets will allow students to research or reference measurements drawn from topics of interest. Rulers, tape measures, meter sticks, and trundle wheels can help students model a measurement or make physical measurements of other items. Place-value charts could support students in changing measurement units. If students choose measurements that interest them but involve computations they are not familiar with yet, they could use a calculator to help them.” Teacher Experience Guide, Conversation Starters states, “When you change a measurement in meters to centimeters, will the numbers get bigger or smaller? Why? When you change a measurement in miles to feet, will the numbers get bigger or smaller? Why?”

An example in Grade 5 includes:

• Topic 8: Geometry, Lesson 3, Student Experience Book, students explore relationships among coordinates and how defining attributes of shapes relate to the coordinates of their vertices. Student Experience Book, Action Task states, “Imagine that this is a map of your community, and your home is at (0, 0). Follow each set of instructions below to see where you might have gone. Think of each move to the right or up as moving 1 street. 1. You and your mom went to your friend’s house and walked twice as many streets to the right as you went up. What coordinates did you land on? 2. You and your dad started at your home and walked 4 more streets to the right than you went up. Where did you go? 3. You and your older brother started at your home and walked some to the right and some up. You walked a total of 10 streets. Where did you go? 4. A trapezoid has vertices at (10, 4), (10, 6), and (12, 5). 5. A square has vertices at (6, 9) and (7, 10). 6. An isosceles triangle has vertices at (6, 2) and (8, 8). 7. A parallelogram has vertices at (9, 12) and (10, 15).” Teacher Experience Guide, Action states, “Students must be accurate when using the provided information.” Teacher Experience Guide, Conversation Starters states, “Which coordinate will be greater―the first one or the second one, or is it not clear? What do you have to make sure of so that it is a trapezoid? Is there more than one possibility for the other vertices?”

The materials reviewed for Experience Math Grades 3 through 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 3-5 grade band engage with MP7 throughout the year. MP7 is found in the three-part lesson structure as students look for and make use of structure to identify patterns and mathematical relationships. Action Tasks provide opportunities for students to recognize regularities in repeated calculations, geometric configurations, and algebraic expressions that support problem solving. The Student Experience Book highlights Essential Understandings that help students connect new ideas to broader mathematical structures. During the Consolidate phase, students share strategies and explain how patterns or structures informed their reasoning. Teacher supports, including Dr. Marian Small’s coaching videos, the Teacher Assistance Panel, “And the Point Is” sections, and “Probing and Extending” questions, guide students to deepen their reasoning and make connections between mathematical ideas.

Across the grades, students engage in tasks that support key components of MP7. These include looking for and describing patterns, identifying structure within mathematical representations, and decomposing complex problems into simpler, more manageable parts. Students are encouraged to analyze problems for underlying structure and to consider multiple solution strategies. They make generalizations based on repeated reasoning and use those generalizations to solve problems efficiently. Teachers support this development by selecting tasks that highlight mathematical structure and by prompting students to attend to and describe patterns they notice. Sessions provide opportunities for students to compare approaches, justify their reasoning, and reflect on how structure helps deepen their understanding. Teachers are encouraged to facilitate discussions and structure lessons in ways that promote the recognition and use of patterns and structure in problem-solving.

An example in Grade 3 includes:

• Topic 10: Adding and Subtracting Whole Numbers, Lesson 2, Student Experience Book, students use properties of operations to create alternative addition or subtraction equations with the same result and show alternative strategies for adding and subtracting. Student Experience Book, Action Task states, “For Questions 1 to 5, fill in the blanks to make each statement true. Do this 2 times for each question. Then, explain how you know your statement is true without calculating the answers. 1. 528 + 123 = \rule{0.5cm}{0.15mm} + \rule{0.5cm}{0.15mm} 2. 417 + 385 = \rule{0.5cm}{0.15mm} + \rule{0.5cm}{0.15mm} + \rule{0.5cm}{0.15mm} 3. 528 + 123 is 250 less than \rule{0.5cm}{0.15mm} + \rule{0.5cm}{0.15mm} 4. 401 - 176 = \rule{0.5cm}{0.15mm} - \rule{0.5cm}{0.15mm} 5. 711 - 135 = \rule{0.5cm}{0.15mm} - \rule{0.5cm}{0.15mm} - \rule{0.5cm}{0.15mm} . For Questions 6 and 7, show or explain 2 strategies for each calculation. 6. 467 + 389 7. 502 - 167.” Teacher Experience Guide, Action states, “Students use properties of addition and subtraction to develop computational strategies that make sense.” Teacher Experience Guide, And the Point Is… states, “This Action Task focuses more on the properties of the operations than only the calculations since we want to emphasize for students how important and useful these properties are … provides an opportunity for students to extend their repertoire of strategies for adding and subtracting … helps students see that there are relationships between calculations that can be explored without getting answers.” Teacher Experience Guide, Conversation Starters states, “What do you know about other sums that are the same as 3 + 4? How could that help? What do you know about other differences that are the same as 20 - 7? How could that help? Would it help to start with the same number or not?”

An example in Grade 4 includes:

• Topic 2: Representing, Comparing, and Ordering Fractions, Lesson 3, Student Experience Book, students use structure to represent fractions greater than 1 and mixed numbers. Student Experience Book, Action Task states, “1. Fill in the blanks to create two fractions greater than 1 and two mixed numbers. Use all the numbers shown. 2. Model the four numbers from Question 1. Use a different type of model for each. Use at least one number-line model. Make sure each model clearly shows what the whole is. 3. Describe two or more things that each model you made for Question 2 shows about the number. 4. Use the fractions from Question 1. Write each fraction greater than 1 as a mixed number. How do you know your answers are correct? 5. Use the fractions from Question 1. Write each mixed number as a fraction greater than 1. How do you know your answers are correct?” Teacher Experience Guide, Action states, “Students begin to notice the relationship between the mixed number and fraction name for a value.” Teacher Experience Guide, In This Task… states, “Students create and model both fractions greater than 1 and mixed numbers, explain what their models show, and then write the fractions as mixed numbers and the mixed numbers as fractions.” Teacher Experience Guide, And the Point Is… states, “This Action Task is designed to focus on representing mixed numbers and fractions greater than 1 using a variety of models … explore the relationship between mixed numbers and fractions greater than 1 … To fill in the blanks for the mixed numbers, students need to realize that the fraction parts must have denominators greater than 2 and 3 respectively. Also, to fill in the blanks for the fractions greater than 1, students need to realize that the numerators must be greater than the denominators.” Teacher Experience Guide, Conversation Starters states, “What models could you use to show fractions? Why are you more likely to write a certain mixed number as 4\frac{1}{8} than as 3\frac{9}{8}? Can you tell from your model what whole numbers your fraction is between?”

An example in Grade 5 includes:

• Topic 11: Rounding, Estimating, and Comparing Decimals, Lesson 2, Student Experience Book, students use structure to compare and order decimal values to the thousandths place. Student Experience Book, Action Task states, “1. Look at the racing speeds in the picture. a. Choose digits for the blanks. b. Compare some of the speeds using > or <. c. Order the speeds from least to greatest. d. Explain how you know which value is greatest. 2. Did the order of the speeds depend on what you put in the blanks? Why or why not? 3. Look at the racing car speeds in the picture again. a. Choose different digits for the blanks so that the order changes. Order the new speeds from least to greatest. b. Choose two of your closest values. Explain how you knew which was greater. 4. Make up speeds for Car G, Car H, and Car I as described below. Use decimals to thousandths. a. Car G is faster than Car B but slower than Car D. b. Car H is a little faster than Car A. c. Car I is almost as fast as Car F. 5. Write two of the speeds from Question 4 in expanded form. Compare the two speeds using < or >.” Teacher Experience Guide, Action states, “Students begin to see that working from the left place-value columns is the way to compare decimal values.” Teacher Experience Guide, In This Task… states, “Students choose digits to complete numbers with decimals to thousandths, and then compare the numbers in pairs to help order them.” Teacher Experience Guide, And the Point Is… states, “This Action Task uses many close values which allows students to see that sometimes we need to compare only the whole-number parts, sometimes we need to compare the tenths digits as well, and sometimes we need to compare the hundredths digits or the thousandths digits … gives students some choice in filling in the digits so they can think more fully about which digits matter and which digits don’t when comparing decimal numbers.” Teacher Experience Guide, Conversation Starters states, “Are there any comparisons you are sure of no matter what goes in the blanks? Could Car B be faster than Car F? Could it be slower? What would make it faster than B, but slower than E?”

The materials reviewed for Experience Math Grades 3 through 5 meet expectations for supporting the intentional development of MP8: Look for and express regularity in repeated reasoning, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 3-5 grade band engage with MP8 throughout the year. MP8 is found in the three-part lesson structure as students they look for and express regularity in repeated reasoning. Action Tasks provide opportunities for students to recognize patterns in calculations and mathematical structures, then apply these observations to develop general methods and efficient solution strategies. The Student Experience Book highlights Essential Understandings that connect new learning to broader mathematical concepts. During the Consolidate phase, students reflect on their processes and explain generalizable strategies through sharing routines such as Math Congress and Student Sharing. Teacher supports, including Dr. Marian Small’s coaching videos, the Teacher Assistance Panel, “And the Point Is” sections, and “Probing and Extending” questions, guide students to analyze repetitions, explain why patterns occur, and evaluate the reasonableness of their results.

Across the grades, students engage in tasks that support key components of MP8. These include noticing and using repeated reasoning to make sense of problems, recognizing patterns, and developing efficient, mathematically sound strategies. Students are encouraged to create, describe, and explain general methods, formulas, or processes based on patterns they identify. Lessons provide opportunities for students to evaluate the reasonableness of their answers and refine their approaches through discussion and reflection. Teachers support this development by structuring tasks that highlight repeated reasoning, prompting students to make generalizations, and guiding them to build conceptual understanding, distinct from relying on memorized tricks. Instructional guidance also encourages teachers to model and elicit strategies that build toward formal algorithms or representations through consistent reasoning.

An example in Grade 3 includes:

• Topic 11: Patterns in Tables and Charts, Lesson 1, Student Experience Book, students explain and generalize patterns in the addition table. Student Experience Book, Action Task states, “For Questions 1 to 4, look at the addition tables. Describe each pattern and explain why each pattern makes sense. … 5. Tell 2 or 3 other patterns you notice in the addition table.” Teacher Experience Guide, Action states, “Students use generalizations that explain the patterns.” Teacher Experience Guide, In This Task states, “Students explain a variety of patterns in the addition table, whether the pattern is pointed out to them or they notice it themselves.” Teacher Experience Guide, And the Point Is… states, “This Action Task includes patterns based on addition and subtraction properties … helps students see that properties of operations can be represented visually … addresses both the structure of the addition table and the properties of the operations.” Teacher Experience Guide, Conversation Starters states, “What do you notice about the numbers in the row and column? What is the same about all the highlighted numbers? Why does it make sense that the numbers go up by 2?”

An example in Grade 4 includes:

• Topic 6: Patterns, Lesson 3, Student Experience Book, students compare two patterns that grow in different ways, one multiplicatively and one additively. Student Experience Book, Action Task states, “For two weeks, Krissy and Alexis collect bread tags. Krissy starts with 1 bread tag on the first day and doubles the number of bread tags each day. Alexis starts with 10 bread tags on the first day and collects 12 more tags each day. 1. Who do you think will have more bread tags at the end of two weeks, Krissy or Alexis? Justify your prediction. 2. Make a table of values to show how many bread tags each child will have each day for the two-week period. 3. Look at the two patterns. a. In that two-week period, which child collects more tags? b. Why do you think that happens? 4. Look at the two patterns. a. Describe something that is true only about Krissy’s pattern. b. Describe something that is true only about Alexis’ pattern. c. Describe something that is true about both patterns”. Teacher Experience Guide, Action states, “Students use patterns related to multiplication and addition to continue the patterns and have a sense of how they grow.” Teacher Experience Guide, In This Task states, “Students compare two patterns that grow in different ways, one multiplicatively and one additively. They make predictions based on pattern rules, then apply the pattern rules to check their predictions.” Teacher Experience Guide, And the Point Is… states, “This Action Task is designed to allow students to use calculators in order to focus on predicting and describing the patterns instead of focusing on the mathematical calculations … help students become aware of the difference in growth rates between patterns based on multiplying by whole numbers greater than 1 and patterns based on adding whole numbers … help students realize that you can’t just look at numbers and know what will happen in a pattern without considering how the numbers will be used.” Teacher Experience Guide, Conversation Starters states, “Who will have more tags on the 3rd day? Does the difference between the number of each girl’s tags stay the same or change?”

An example in Grade 5 includes:

• Topic 13: Multiplying and Dividing with Decimals, Lesson 1, Student Experience Book, students create additional number patterns by multiplying and dividing by powers of 10 and examine how the patterns relate. Student Experience Book, Action Task states, “1. Choose a whole number less than 100 to multiply by 10^4, then 10^3, then 10^2, and so on, to make a pattern. Continue the pattern until decimals are involved. 2. Use the whole number you started with in Question 1. Make a pattern based on dividing the number by 1, then 10, and then 100. 3. How might you use a place-value chart to show why it makes sense that you got the decimal numbers that you did when you divided by 10 and 100? 4. What did you notice about your two patterns in Questions 1 and 2? Why does that make sense? 5. What do you think might be another way to write 0.01\times583? Why?”. Teacher Experience Guide, Action states, “Students notice the same digits and the movement of the digits as they multiply or divide by increasing powers of 10.” Teacher Experience Guide, In This Task states, “Students create additional number patterns by multiplying and dividing by powers of 10, and they look at how the patterns relate.” Teacher Experience Guide, And the Point Is… states, “This Action Task is designed to ensure that students see that the type of patterns they saw in the Minds On Activity occur no matter what number is selected.” Teacher Experience Guide, Conversation Starters states, “Will the answers increase or decrease? Why? What do you notice about the digits in the answers? Do you think a similar pattern will happen if you begin with a three-digit number instead of a two-digit number?”