The materials reviewed for STEMscopes Math Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

STEMscopes materials develop conceptual understanding throughout the grade level. In the Teacher Toolbox, STEMscopes Math Philosophy, Elementary, Conceptual Understanding and Number Sense, STEMscopes Math Elements, this is demonstrated.  “In order to reason mathematically, students must understand why different representations and processes work.” Examples include:

• Scope 4: Represent and Compare Decimals, Explore 3: Represent and Compare Decimals, Procedure and Facilitation Points, develops students’ conceptual understanding. (Sample answers follow some questions). “Discuss what students may remember about comparing fractional amounts. Use the following scenario to help them think about the need for the whole to be the same when comparing fractional parts. Let’s say your mom decided to stop at the bakery on the way home from work. She wanted to reward you and your sibling with some sweet treats to celebrate your excellent grades. She hands you 0.50 of a cupcake, and you are so excited! You look over and notice that she got your sibling 0.50 of an entire cake. DOK-2 Is this an equal amount? Why or why not? DOK-2 What is similar about the sweets you both received? DOK-1 So, to make a comparison between two things, what has to be the same? … Read this scenario: Two amusement parks are interested in opening up in your area of town. However, the city says that only one of the parks will be built. Both parks gave your city officials some statistics about their parks in an effort to be chosen. The city has decided to put it to a vote. Tell students they will be comparing features of both parks and deciding which park they want in their city based on the statistics. Distribute a Student Journal to each student. Give each group a copy of the Amusement Park Comparison. Discuss the information. Have students use their base ten blocks to show the distance from the City Center to Splashing Wild on the first row of their Place Value Mat and the distance from the City Center to Screaming Good Time on the second row. Be sure to check that students are correctly using the base ten blocks to build the number. One whole can be represented using a flat. Have students use the dry erase markers to write each digit in each place value on the mat. Have students shade in a representation of the base ten blocks on the given grids for both distances in their Student Journal. Have students use the Place Value Mat, folded on the dotted line, to compare the numbers. They should look at the wholes first and relate it to the model they built. Discuss the following: DOK-1 Which number is greater? Can you tell? Explain. DOK-1 What process could you follow to compare numbers? Next, have students uncover the tenths and hundredths places on the Place Value Mat and relate it to the model they built. DOK-1 Which park is farther away? DOK-2 How do you know? DOK-1 How could we record this comparison using symbols? 2.46 > 2.36.Is there another way we could write it? DOK-1 How could we read this statement? Draw the following number line on the board, and ask students where these two numbers would fall on the number line.Have students record the number comparison using the appropriate symbol on their Student Journal page. Have students compare the rest of the statistics using the same process described above. Have students build each number (as needed), create their visual model, and record the digit in each place value on the mat. Students should fold the Place Value Mat and look at wholes first. If the wholes are the same, then they should uncover the decimal values to reason through which is greater or less. Then have students record their work and comparison statements on their Student Journal page. Students should use their visual models as support for their reasoning. The goal is for students to be able to compare two numbers without the use of base ten blocks. However, allow students to use the blocks if they are still needed. After the Explore, invite the class to a Math Chat to share their observations and learning.” (4.NF.7)

• Scope 6: Compose and Decompose Fractions and Mixed Numbers, Explore, Explore 1– Compose and Decompose Unit Fractions, Procedure and Facilitation Points, students develop conceptual understanding of composing and decomposing fractions using unit fractions. “Read the following scenario: Ann’s Bakery Shop is having a grand opening! They have baked several different pies and are handing out free samples at various serving stations. At each station, you will determine how many slices of each type of pie was handed out at the grand opening. You will use this information to see how much pie was given out. 2. Each group will start at a different serving station. 3. Give students about 10 minutes at each station before rotating. 4. When students begin, they will complete the following steps at each station: a. Read how many slices of pie were handed out at that station and pull that many slices of pie out of the bag. b. Determine how many slices make up a whole pie of that flavor in order to determine the denominator for the fractional part of each piece. c. Write a number sentence to show the sum of the fractional parts of each piece that was handed out. d. Assemble the pie pieces into as many whole pies as possible, Dray a model of the slices on the Student Journal. e. Write a mixed number representing how many slices of that flavor were handed out. 5. As students are working cooperatively, use the following guiding questions to extend their thinking. a. DOK-1 How many pieces are in one whole ___ pie? b. DOK-1 What is the fractional part of one slice of ___ pie? c. DOK-1 How many slices of ___ pie were handed out? d. DOK-1 How could we find the fractional part of the number of slices of ___ pie that were handed out? e. DOK-1 What do you notice about the fractional sum? (Explain that this is called an improper fraction.)  f. DOK-1 How many whole pies can you make with the number of slices of ___ pie that were handed out? g. DOK-1 How much of the next pie do we have? (Model for students how this can be recorded as a mixed number.) 6. After completing each station, students will complete the reflection questions on the last page of their Student Journal. 7. After the Explore, invite the class to a Math Chat to share their observations and learning. 8. When students are done, have them complete the Exit Ticket to formatively assess their understanding of the concept.” (4.NF.3)

• Scope 8: Equivalent Fractions, Explore, Skills Basics–Identity Property of Multiplication, Procedure and Facilitation Points, students develop conceptual understanding of multiple and common denominators. “1. Discuss the following questions: a. Have you ever dressed up in a costume? b. Did it change who you actually are? 2. Explain that putting on different clothes or costumes doesn’t change students’ identities, which is who they really are. Numbers can have identities, too. They stay the same when multiplied by a certain factor. 3. Instruct students to draw a large circle on their Identity Property Work Mats and place 8 counters in the circle. Discuss the following questions: a. How many groups do we have? b. How many counters do we have in the group? c. How many total counters are there? d. How do we write this model as a multiplication equation? e. Did the 8 keep its identity — did the 8 stay an 8? 4. Tell students to remove the counters from the circle and erase their Identity Property Work Mats. 5. Instruct students to draw two large circles on their Identity Property Work Mats and place 8 counters in each circle. Ask the following questions: a. How many groups do we have? b. How many counters do we have in each group? c. How many total counters are there? d. How do we write this as a multiplication equation? e. Did the 8 keep its identity — did the 8 stay an 8? 7. Ask the following question: a. When does a number keep its identity? When it is multiplied by 1, 8. Explain that this is called the Identity Property of Multiplication. This property states that any number multiplied by a factor of 1 keeps its identity because it stays the same. 9. Discuss with students the relationship between the Identity Property of Multiplication and identifying and generating equivalent fractions. Discuss the following question: a. How does the Identity Property of Multiplication and what you know about factors and multiples help you understand equivalent fractions? When you multiply a fraction by a fraction with the same multiple in the numerator and denominator, like $\frac{2}{2}$, you are actually multiplying the fraction by a factor of 1 whole. The equivalent fraction’s identity, or amount, does not change, but the size and number of pieces do. 10. Distribute the Student Handout. Have students identify and use the Identity Property of Multiplication.” (4.NF.1)

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

• Scope 5: Represent and Compare Decimals, Explain, Show What You Know–Part 1: Decimal Notation for Denominators of 10, students write fractions with a denominator of ten in decimal notation for fractions with the denominator of 10. Students should individually complete the Show What You Know activity that correlates with the Explore activity they just completed. Each Show What You Know piece correlates with the same number Explore. For example, Show What you Know Part 1 will allow students to practice the skills they developed in Explore 1. “Complete the missing information. Value (words) Nine-tenths. Shaded Base-Ten Model, Fraction Notation, Decimal Notation.” (4.NF.6) 

• Scope 9: Add and Subtract Fractions and Mixed Numbers, Explore 3: Subtract Fractions and Mixed Numbers with Like Denominators, Essential Cards, Part 1, students solve tasks involving fractions, using manipulatives. “Water: Lily brought $2\frac{3}{5}$ liters of water on her hiking trip. During the hike, she drank $\frac{4}{5}$ liters. How many liters does Lily have left?” Question two: “Trail Mix: Lily brought $2\frac{1}{4}$ cups of trail mix on her hiking trip. During the hike, she ate $1\frac{2}{4}$ cups of trail mix. How many cups of trail mix does Lily have left?” (4.NF.3c)

• Scope 14: Division Models and Strategies, Show What You Know–Part 1: Sharing Equally, students solve whole-number division problems. Examples include: “For each situation, show how you solve the problem using base ten blocks, write an equation to represent the problem, and write a solution statement.” Question one: “At the start of the school year, there are 230 pencils that will be shared equally among 5 classrooms. How many pencils will each classroom get?” Space is provided and labeled “Base-Ten Model, Equation, Solution Statement”. Question two: “There are 6 coaches at a basketball clinic. Each coach needs to work with the same number of players. There are 144 players registered for the clinic. How many players will each coach work with?” Space is provided and labeled “Base-Ten Model, Equation, Solution Statement”. Question three: “The cafeteria has 8-ounce juice cups. How many servings can they make for the day if they have a total of 576 ounces of juice?” Space is provided and labeled: “Base-Ten Model, Equation, Solution Statement.” (4.NBT.6)

The materials reviewed for STEMscopes Math Grade 4 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

STEMscopes materials develop procedural skills and fluency throughout the grade level. In the Teacher Toolbox, STEMscopes Math Philosophy, Elementary, Computational Fluency, STEMscopes Math Elements, these are demonstrated.  “In each practice opportunity, students have the flexibility to use different processes and strategies to reach a solution. Students will develop fluency as they become more efficient and accurate in solving problems.” Examples include:

• Daily Numeracy: Fourth Grade, Activities, Daily Numeracy–Solve It, Procedure and Facilitation Points, and Slideshow, engages students in subtracting multi-digit numbers using various strategies. Slide 3, "$9,323-8,999$" “2. Display the slideshow prompt of the day, and ask students to silently think and solve. Instruct students to give hand signals when they are ready to answer. 3. Call on students to give out answers only. Record student answers on chart paper. 4. Ask students to volunteer and to explain the strategies they used to get answers.” … “5. As students share strategies, ask the class if they agree or disagree, and provide sentence stems for their responses. a. I agree because…; b. I disagree because…; c. Can you explain why you …?; d. I noticed that…; e. Could you…?” (4.NBT.4)

• Scope 4: Addition and Subtraction Algorithms, Explore 1: Multi-digit Addition, Procedure and Facilitation Points, develops students’ procedural skill and fluency as they perform multi-digit addition and subtraction. “Read the following scenario. ‘You hit the lottery! You have decided to use your winnings to travel to new places around the world. Grab your passport and get ready to see new sights!Invite students to look at the Passport Pursuit cards so they can see the beautiful places they will be visiting over the next year.’ Have students pull out their boarding passes and have them match up the connecting flights for Trip 1. The different trips are listed in the Student Journal. Once they have chosen the correct boarding passes for Trip 1, have the students match the correct Passport Pursuit cards for this trip to their boarding passes so they can get an idea of the locations they will be visiting. Students will then work as a group to take the distances traveled for each boarding pass and add them together to find the total distance traveled for that trip.Instruct students to work together using their addition work mats to add their numbers using multiple strategies, including an open number line, partial sums, and the standard algorithm. Students can either record their open number line or partial sums strategy on their Student Journal along with the standard algorithm. If students are struggling with the algorithm, allow them to use the work mat and place value disks to model regrouping. After the students have completed solving for Trip 1, ask the following questions to check for understanding. ‘DOK-3 Which addition strategy did you find to be the most efficient way to solve this problem? DOK-1 How do you solve using the standard algorithm? DOK-1 How do we regroup in Trip 1 using the standard algorithm? DOK-2 Which addition strategy had a similar way of regrouping as the standard algorithm? DOK-2 Did you find that all of the strategies have regrouping?’ On the last page of the Student Journal, encourage students to find the total distance of all 3 trips on their own. After the groups have had enough time to complete their work, invite students to gather together to share observations and learning in a math chat. After the Explore, invite the class to a Math Chat to share their observations and learning.” (4.NBT.4)

• Scope 17: Problem Solve Using the Four Operations, Explore, Explore 2–Problem Solve Using the Four Operations (Level 1), Procedure and Facilitation Points, engages students in procedural fluency as they write equations using the four operations with an unknown variable. “9. Do a few quick examples as a class. Tell students they will be writing equations using letters on their dry erase boards as practice. Students should write each equation on their board and then share and discuss with their group. (Remind them that they don’t need to solve the equations right now. Just build them.) a. DOK-1 The number 247 minus 43 equals some number. b. DOK-1 The number 34 times some number equals 612. c. DOK-1 Some number divided by 28 equals 4. d. DOK-1 The number 1,200 plus some number equals 1,399. 10. Have students go back and replace unknowns in their equations with letters to replace the unknowns.” (4.OA.3)

The materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:

• Scope 3: Rounding, Elaborate, Fluency Builder–As Close to 100,000 as You Can, Instruction Sheet, students build procedural fluency as they round numbers based on place value. “3. Each player draws one card from each digit set to form a number. For example, if you draw a 1 from the ones cards, a 2 from the tens cards, a 3 from the hundreds cards, a 4 from the thousands cards, and a 5 from the ten thousands cards, the player’s number is 54,321. Each player writes his or her number on the student recording sheet. 4. A place value card is drawn (players alternate drawing the place value card between each round). Players round their number to the place value position indicated on the card and record it on the student recording sheet. For example, if a player formed the number 54,321 and the “Round to the nearest thousands” place value card was drawn, the player would round the number to 54,000. Each player’s score is the rounded value for that round.” (4.NBT.3)

• Scope 4: Addition and Subtraction Algorithms, Evaluate, Skills Quiz, students demonstrate  procedural skill and fluency of adding and subtracting multi-digit whole numbers using the standard algorithm. Question 1: "$6,037+4,510$" Question 2:  "$3,006+2,872$" Question 3: "$8,512+4,284$" Question 4: "$6,199+4,832=$" Question 5: "$4,622+1,307$" (4.NBT.4)

• Scope 15: Prime and Composite Numbers, Elaborate, Fluency Builder - Multiple Match, Instruction Sheet, students engage in procedural fluency as they match a multiple with its composite. “1. Shuffle the cards, and place them face down to form a $4\times6$ array. 2. The first player flips over two cards to try to find a match. 3. If the player finds a factor and its composite match, the player keeps the matched set. 4. If the player does not find a match, he or she places the turned cards face down again, and it is the next player’s turn. 5. If the player finds a prime number, the player keeps the card, but his or her turn is over.” (4.OA.4)

The materials reviewed for STEMscopes Math Kindergarten meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics.

STEMscopes materials include multiple routine and non-routine applications of mathematics throughout the grade level, both with teacher support and independently. Within the Teacher Toolbox, under STEMscopes Math Philosophy, Elementary, Computational Fluency, Research Summaries and Excerpt, it states, “One of the major issues within mathematics classrooms is the disconnect between performing procedural skills and knowing when to use them in everyday situations. Students should develop a deeper understanding of mathematics in order to reason through a situation, collect the necessary information, and use the mechanics of math to develop a reasonable answer. Providing multiple experiences within real-world contexts can help students see when certain skills are useful.” 

This Math Story activity includes both routine and non-routine examples of engaging applications of mathematics. For example:  

• Scope 10: Multiply Fractions by Whole Numbers, Elaborate, Math Story–Aspen’s Birthday Bash, students solve both routine and non-routine problems with teacher support. “Read the passage and answer the questions that follow.  1. How many bottles of ginger ale would be needed when Aspen asked the girls to triple the recipe? A 2, B 1, C 1, D 3, 3. Which equation correctly determines how many pizzas Aspen needed to order if she wanted enough for the girls at her party and 2 extras for her family? A $(\frac{1}{2}\times8)-2$, B $(\frac{1}{2}\times8)+2$, C $(2\times8)+2$, D $(4\times8)+2$” (4.NF.4c)

Engaging routine applications of mathematics include:

• Scope 2: Place Value of Whole Numbers, Explore 1–Place Value Relationships, Procedure and Facilitation Points, students develop application through a routine problem with teacher support as they collect donations for a disaster relief organization for 8 days and find the place value relationships for each day based on a company’s donation match. Amounts will build up to one million. “1. Distribute a Student Journal to each student. 2. Distribute a set of place value disks to each group along with the ones and thousands period Place Value Charts. Do not pass out the millions yet. Students should place the charts side by side so that the ones period is on their right and the thousands period is on the left. 3. Allow students to open the bag and sort the place value disks and place above their Place Value Charts. Students will notice the million disk and start to wonder where it goes. Distribute the millions period chart, and ask students what place value that disk is. 4. Read the following scenario to students: a. A local disaster relief organization is having its annual Penny Palooza! The Penny Palooza is an 8-day fundraiser in which donors give different amounts of money in the form of pennies to support disaster relief. The Money Matchers Company has graciously agreed to match the donation amount by also donating ten times the amount of pennies that Penny Paloozwa receives daily. 5. Distribute a set of Scenario Cards to each group. 6. Students will read the information on each Scenario Card for each day and work with their group to solve what the match amount from the Money matchers Company should be. 7. Students should use the place value disks to build the donation amount and place it on their Place Value Chart. They should then use the place value disks (as needed) to build ten times that amount (matched by the Money Matchers Company) and figure out how the value changes for each digit. 8. Students should record the amount of the donation they received for that day on their Student Journal page, as well as the donation from the Money matchers Company. For each day, they should write one multiplication equation and one division equation showing the relationship between the amount collected and the donation from the Money Matchers Company. a. If students begin to notice the pattern of what happens to the digits when multiplying by ten, they may no longer need to build the model. Just make sure they can explain their thinking. 9. While monitoring students, ask them engaging questions such as the following:  a. DOK-1 What do you notice about the digits in the number when you multiply it by ten? b. DOK-2 If you have the same number of disks in the next value, what is the same between the two numbers, and what is different? c. DOK-2 If you were moving from a higher place value to a lower place value, what operation would you use? What would this do to the value?...” (4.NBT.1)

• Scope 13: Multiplication Models and Strategies, Explore 2–Multiply Four-Digit by One-Digit Numbers: Area Models, Procedure and Facilitation Points, students develop application with teacher support as they use what they have learned about arrays to create area models to multiply numbers up to four digits by one-digit numbers. “1. Give each group a clear sheet protector with the white paper inside. They will also need a dry-erase marker, eraser, and a set of Place Value Cards. Give each student a Student Journal. 2. Read the following scenario. Illumination Theater is an amphitheater that is used to entertain large groups. The company has several amphitheaters around the state. Each theater is used to hold concerts, plays, symphonies, etc. The theater has a certain number of rows, and each row has a certain number of seats in it. We need to figure out how many total seats there are in each theater! 3. Tell students to build an array to model the rows and seats in each row for the first theater. The first theater has 5 rows with 6,298 seats in each row. They will build their model on top of the clear sheet protector. They will then use a dry-erase marker to trace all the cards in their array. Have them draw lines between the hundreds, tens, and ones. Students can lay the blank, covered paper end so the arrays will fit. 4. Tell students to remove their cards. They have now created their area model! 5. Allow students to label the length and width of each rectangle of the area model. Explain that when using this model, you are finding the area (the space inside the rectangle) of each rectangle. 6. Have students find the total area of each piece of their model and record if using an equation. These equations will be written inside their area models. They should circle the products. Finally, they will add the total from each piece of their model to find the final product. 7. Discuss the following: a. DOK-2 What did we do to the 628 in order to multiply it by 5? 8. Explain that when we do this, we are using what is called the distributive property. The distributive property allows us to multiply one chunk at a time, just like students did with 628 times 5. 9. Explain that there is a special way we can record the equations to show how we multiplied the numbers. We can use parentheses to show each part of our model. On the board, write the equation that shows the distributive property for this model: $(5\times600)+(5\times20)+(5\times8)$. Ask the following question. a. DOK-1 How can we now find the total?...” (4.NBT.5)

Engaging non-routine applications of mathematics include:

• Scope 3: Rounding, Explore 1–Money in the Bank!, Procedure and Facilitation Points, students develop applications through a non-routine problem with teacher support as they place a number on a number line between intervals of 10, 100, 1000, 10,000, or 1,000,000. Students will then use relative language to describe the position of the number between two intervals in order to round whole numbers. “Part I, 1.Have students gather around the number line with multiples of 1,000. Ask the following question: DOK-1 What multiples does this number line show? 2. Read the following scenario: You and your group have started a toy business. The number on which your beanbag lands is the amount you will report as your estimated savings each week from the sales of your toys. You need an estimate of your business’s profit - how much money you are able to save each week–in order to be able to buy more materials to make new toys. 3. Ask the following questions: DOK-2 When we want an estimate, what can we do? DOK-2 When is rounding numbers useful? 4. Choose one student from each group to demonstrate how to toss the beanbag onto the 1,000 number line. 5. Demonstrate how to toss the beanbag between the numbers on the number line. Students should take turns standing at the 0 mark and tossing the beanbag so that it lands somewhere along the number line (profit). 6. Have students determine the number that shows the location of the beanbag their teammate tossed. 7. Students will then determine which amount is closest to that location and record it on their Student Journal as their estimated savings. 8. Next, repeat by asking students to take turns in their groups at the multiples of 1,000 number line. They will record their experience on the second page of their Student Journal and discuss the questions in their group 9. Assist students in the discovery of the concept as they throw their beanbags. DOK-1 Describe the location of the beanbag along the number line. DOK-1 Which numbers is your bean bag between? DOK-1 Is your bean bag closer to 200 or 300? Encourage students to use language that helps them understand the relative distance, such as closer, nearer, almost, and farther. Part II, 1. Have students rotate to the other five number lines with their groups. They will write down their estimated numbers on their tables and begin rounding in their groups. 2. As students are completing the activities at each number line, have the following math chat to help students make sense of using the number line to round numbers. 3. After the Explore, invite the class to a Math Chat to share their observations and learning. 4. Overall, the students’ answers should reflect their understanding of looking at the digit in the place value their number line is counting by. Then, they should look at the digit to the right to see which multiple it is closest to.” (4.NBT.3)

• Scope 14: Division Models and Strategies, Explore 1–Sharing Equally, Exit Ticket, students independently demonstrate application through a non-routine problem as they model division of larger numbers, using base ten blocks and generic school supplies. “Divide the amount below evenly between six groups. Write an equation at the bottom of the page to represent your work. (2 thousands, 5 hundreds, 2 tens, 6 ones)” (4.NBT.6)

The materials reviewed for STEMscopes Math Grade 4 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout the grade. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, and application include:

• Daily Numeracy: Fourth Grade, Activities, Daily Numeracy–Guess The Number, Slideshow, Range 4 and Number 4, and Procedure and Facilitation Points, students develop procedural fluency of numbers and place value to identify a given number. Two slides are used for each Daily Numeracy activity. Range 4, “Between 750 and 850” Number 4, “805”. Procedure and Facilitation Points, “1. Gather students in a group with a piece of chart paper and a marker. Students should not have anything with them for this activity. 2. According to the number range on the prompt, allow students to ask yes/no questions to help guess the number. 3. Respond to students with “yes” or “no,” according to the number given on the prompt. Record student questions and guesses on the chart paper so students can see what others have asked. 4. Possible student questions: a. Is the number odd? Is the number even? b. Is the number greater than or less than  ___? c. Is the number between ___ and ___? d. Does it have a (digit) in the (hundreds, tens, ones) place? e. Does it have ___ digits? 5. When students have guessed the number, plan to project the slideshow prompt and to discuss by using relevant guided questions. a. What questions were the most helpful when guessing the number? b. How did you eliminate other numbers?” (4.NBT.1)

• Scope 12: Multiplication Models and Strategies, Explain, Show What You Know–Part 1: Multiply Four-Digit by One-Digit Numbers Arrays, students show conceptual understanding of multiplication of numbers up to four digits by a one-digit number, using multiples of 10 and arrays. Students should individually complete the Show What You Know activity that correlates with the Explore activity they just completed. Each Show What You Know piece correlates with the same number Explore. For example, Show What You Know Part 1 will allow students to practice the skills they developed in Explore 1. “Create an array for the following: $3.312\times6=n$ Write equations for the following: $3,312\times6=n$ Equation to represent thousands:___, Equation to represent hundreds: ___, Equation to represent tens: ___, Equations to represent ones: ___, Equation to show partial products sum: ___, n = ___” (4.NBT.5) 

• Scope 19: Angles, Explain, Show What You Know–Part 3: Sketching Angles of Specified Measures, students demonstrate application through a non-routine problem as they draw angles of a certain measurement using a protractor. “Draw a polygon with the following attributes: Exactly 6 angles; Vertices labeled A, B, C, D, E, and F; At least one acute angle; At least one obtuse angle ___ Complete the missing information. Angle Name ∠ABC; ∠BCD; ∠CDE; ∠DEF; ∠EFA; ∠FAB” (4.MD,6)

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

• Scope 2: Place Value of Whole Numbers, Explain, Show What you Know–Part 2: Reading and Writing Multi-Digit Whole Numbers, students show conceptual understanding alongside procedure skill and fluency of writing numbers in base ten numerals, expanded form, and work form and represent them with a model using place value disks. Students should individually complete the Show What you Know activity that correlates with the Explore activity they just completed. Each Show What You Know piece correlates with the same number Explore. For example, Show What You Know Part 1 will allow students to practice the skills they developed in Explore 1. “Fill in the missing information for each number. Numerals 672,781, Number Name ___,  Expanded Form ___, Numerals ___, Number Name one hundred forty-two thousand, five hundred eight,  Expanded Form ___” (4.NBT.1)

• Scope 4: Addition and Subtraction Algorithms, Elaborate, Spiral Review–National Donut Day, Question 3, students demonstrate application of their knowledge of place value alongside procedural fluency as they round numbers to the nearest 1,000. “A bakery in a big city prepared for the celebration by making the following amounts of donuts. Glazed 10,345; Rainbow Sprinkle 1,950; Chocolate 1,500; Jelly-filled 630. Round each number to the nearest thousand. Glazed: ___, Chocolate: ___, Rainbow Sprinkle: ___, Jelly-Filled: ___” (4.NBT.3)

• Scope 13: Multiplication Models and Strategies, Explore, Explore 1, Multiply 1-digit by 4 Digit Numbers: Arrays, Procedure and Facilitation Points, Part 1, develops students’ conceptual understanding alongside procedural fluency as they multiply by 10 fluently. (Sample answers follow some questions). “Give each group a set of base ten blocks and Place Value Cards. Read the following scenario. You are employees at Cra-Z-Crafts, a local craft store. You are going to help with the quarterly inventory. They need your help to figure out how many craft supplies they have in the store. We will start by finding out how much paper they have. There are 3 boxes of paper, and each box contains 123 reams of paper. Encourage students to use the base ten blocks to show how they could figure out the product of 123 and 3. Support students by asking them to think of the problem as “groups of.” Students will show this in various ways such as groups and arrays. DOK-1 What would be a good estimate of our product? Encourage students to think about their multiples of 10 and 100. $120\times3$, $100\times3=300$, or $12\times3=36$ so $120\times3=360$. The answer should be around 300 to 360. Invite a student who used an array to talk through how he or she modeled the problem… Discuss the following. DOK-1 How many are in each row? DOK-1 How many rows are there? DOK-1 So how many hundreds do we have in all? Explain how you know. DOK-1 How could we write this as an equation to show the value of these hundreds? DOK-1 How many groups of tens do we have now? DOK1 How could we write this as an equation to show the value  of these tens? DOK-1 How many ones are in each row? DOK-1 How many rows are there? DOK-1 So how many groups of ones do we have? DOK-1 How could we write this as an equation to show the value of these ones? DOK-1 How could we find the total product? DOK-1 Was this around our estimation? … DOK-1 How many thousands are in each group? DOK-1 How many groups of 2,000 do we have? DOK-1 What equation could we write for 4 groups of 2,000? DOK-1 How many hundreds are in each group?  DOK-1 How many groups of 300 do we have? DOK-1 What equation could we write for 4 groups of 300? DOK-1 How many tens are in each group? DOK-1 How many ones are in each group? DOK-1 How many groups of 5 do we have? DOK-1 What equation could we write for 4 groups of 5? DOK-1 How do we find the total amount? ” (4.NBT.5)

The materials reviewed for STEMscopes Math Grade 4 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with the Math Practices across the year and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the scopes. MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the scopes. Examples include:

• Scope 7: Compare Fractions, Explore, Explore 3–Comparing Fractions Using Benchmarks, Standards of Mathematical Practice, “MP.1 Make sense of problems and persevere in solving them: As students compare two fractions, they will look to see how the number and size of the parts are different as they know the fractions refer to the same size whole. They will persist in trying to determine the best way to solve. Benchmark fractions might be the best way to make sense of the comparison. Finding a common denominator or numerator may be the better choice. Students look closely to make sense of the meaning of the fractions involved, consistently checking themselves, asking ‘Does this make sense?’” In Procedure and Facilitation Points, “Part I, 1. Distribute Student Journals to students. 2. Present students with the following scenario. For the next 10 days, Cecia will participate in the 10-day, $\frac{1}{2}$-mile walk challenge. Her goal is to walk $\frac{1}{2}$ a mile or more each day for 10 days. Use fraction circles to determine whether or not she met her goal. 3. Allow students to use fraction circles to decide whether Cecia met her goal each day. As students are working, ask the following questions. a. DOK-1 How do you know whether or not she met her goal? b. DOK-2 If she met her goal on one day but did not on another, which fraction would be bigger? c. DOK-2 Could I use one-half as a benchmark to compare other fractions? d. DOK-2 Are there other benchmark fractions we could use in order to compare fractions? e. DOK-2 What are some examples of this?”

• Scope 8: Equivalent Fractions, Explore, Explore 2–Modeling Equivalence on a Number Line, Standard and Mathematical Practice, “MP.1 Make sense of problems and persevere in solving them: As students compare two fractions, they will look to see how the number and size of the parts are different as they know the fractions refer to the same size whole.  They will persist in trying to determine the best way to solve. Benchmark fractions might be the best way to make sense of the comparison. Finding a common denominator or numerator may be the better choice. Students look closely to make sense of the meaning of the fraction involved, consistently checking themselves, asking ‘Does this make sense?’”In Procedure and Facilitation Points, “1. Ask students what they have already learned about equivalent fractions. 2. Introduce students to the scenario. You are in charge of planning a color run! The color run is a two-mile race where people enjoy colorful bubbles, powdered paint, music, snacks, and more. It takes a lot of different people to make a color run happen, and you are the one who needs to organize all the volunteers and vendors. You want to make sure everyone is in the right place, so you must be ready to communicate everyone’s location in a variety of ways. 3. Distribute a Number Line Work Mat, dry-erase marker and eraser, colored pencils, and a set of Number Line Spacers to each pair of students. 4. Start with the first detail of the race listed on the Student Journal. Students will use the spacers to partition their number line appropriately and locate the point described on the number line. 5. Challenge students to find all the different ways they can describe the exact location. They should record their findings and sketch their number line model on their Student Journal. Students can use different colored pencils to show the different ways they partitioned their number line. 6. Challenge students to record an equation that proves the fractions they found are equivalent to the original. 7. Students should repeat the same process for each detail and answer the reflection questions at the end. 8. As students are working, monitor their work and check for misconceptions. Use the following engaging questions to support their learning. a. DOK-1 How could you find an equivalent fraction to represent this location? b. DOK-1 How do you know if two fractions are equivalent on a number line? c. Support students in the way they think about the Salty Snacks area. Use guiding questions to help students see that they can use or combine equal groups of pieces to find an equivalent fraction as well. Students should see how division can be used to show how a fraction like $\frac{4}{16}$ is equivalent to $\frac{1}{3}$, $\frac{2}{6}$. etc. 9. After the Explore, invite the class to a Math Chat to share their observations and learning. 10. When students are done, have them complete the Exit Ticket to formatively assess their understanding of the concept.” 

• Debate, students make sense of problems and persevere in solving them as they determine if their answer makes sense and reflect and revise their problem solving strategy. “Granger Elementary School staff was told that they would be getting two new rectangular playground sets added to their recess field! One set would cover 15 square feet and the other set would cover 24 square feet. Luckily, the design of the playground set could be adjusted to best fit the playground needs of the students. However, it was discussed that the playground set that was 15 square feet could only have one design because 15 is an odd number, so it must be prime. The 24-square-foot playground set could have more than one design because 24 is an even number and must be composite. Do you agree with the following? The number 15 is prime and the 15-square-foot playground set can only have one rectangular design. The number 24 is composite and the 24-square-foot playground set can have more than one design. Use the space below to draw out possible playground designs for each set and explain your reasoning.”

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the scopes. Examples include:

• Scope 4: Addition and Subtraction Algorithms, Explore, Explore 3–Adding and Subtracting Strategies, Standards of Mathematical Practice, “MP.2 Reason abstractly and quantitatively: As students solve, they determine the value of the numbers given. Decisions are made depending on the operation needed and the relationship between the numbers. When the meaning of the quantities are understood, a reasonable strategy is chosen to solve.” Procedure and Facilitation Points, “1. Invite the class to play a game. Ask how many people have heard of the game “Would You Rather?” 2. If students have not heard of the game, explain that you are given 2 options and you have to pick the one you would prefer. 3. Play a couple of rounds with the students to get them excited. Questions can be individualized for each class, or use the following examples. Students can discuss with their group. Invite a couple of students to share their answers and why they chose that. a. Would you rather miss recess 1 day but get 10 extra minutes every other day that week, or don’t miss recess but have to play inside? b. Would you rather eat cafeteria food or your favorite vegetable for a week straight? c. Would you rather have an entire day of math or an entire day of reading? d. Would you rather have winter forever including snow or summer forever with 100 degree heat? 4. Explain to students that there is no right or wrong answer, but that they choose the answer based on what they like the most. 5. Pass out the student Journals, addition and subtraction work mats, dry erase markers, and bags of task cards. Read through the directions and have groups start working together to solve. 6. Students will choose two strategies to solve each problem, justify their choice of strategies, and explain which strategy is the most efficient. 7. Allow students to work through each problem with their group using their work mats and then record their answers on their Student Journal…”

• Scope 13: Multiplication Models and Strategies, Explore 4–Area Models, Content Support, “MP.2 Reason abstractly and quantitatively. As problems are worked, students will use various representations and approaches to solve, making connections between the multiplication and the representations. They are able to explain their reasoning using place value language or how they applied one of the properties of operations.” In the Exit Ticket, students apply the area model to a multiplication problem to determine if the restaurant made a record-breaking pizza. Students are given space in a table to make an area model of the problem and identify partial products and then the solution. “Multiplication Area Models Exit Ticket:  Sadly, Paciano’s Pizza Parlor did not win the biggest rectangular pizza competition, so they decided to make a bigger rectangular pizza to see if they could beat the record area of 1,785 square inches. They made a pizza that was 63 inches long and 41 inches wide. Make an area model of Paciano’s pizza, and then write the products for each section. Finally, write the equation that represents the problem and find the total area. See if they were able to create a winning monster pizza! Did Paciano’s make the biggest monster pizza?___” 

• Scope 16: Multiplicative Comparisons in Multiplication and Division, Explore, Explore 1–Model Multiplicative Comparisons, Standards of Mathematical Practice, “MP.2 Reason abstractly and quantitatively: Students recognize additive and multiplicative comparisons. They make sense of the meaning of quantities and write equations with variables to represent an unknown quantity.” Exit Ticket: “Read the scenario below, and create a model to represent your multiplicative comparison. Then complete the sentence stem to explain your model.  Farmer Joe has 35 acres of farmland. This is 5 times more than the acres that Farmer Susan owns. How many acres does Farmer Susan own?”

The materials reviewed for STEMscopes Math Grade 4 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials provide opportunities for student engagement with MP3 that are both connected to the mathematical content of the grade level and fully developed across the grade level. Mathematical practices are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. Students construct viable arguments and critique the reasoning of others, in connection to grade-level content, as they work with support of the teacher and independently throughout the Scopes. Examples include:

• Scope 6: Compose and Decompose Fractions and Mixed Numbers, Explore, Explore 2–Compose and Decompose Fractions in Multiple Ways, Standards for Mathematical Practice, Procedure and Facilitation Points, students build experiences with MP3. In the Standards for Mathematical Practice, the program notes the work with MP3. “MP.3 Construct viable arguments and critique the reasoning of others: Students will make conjectures and explore their solutions, looking for evidence of proof using visual models. Students listen to others, asking clarifying questions and expecting feedback. They may provide counterexamples to justify conclusions.” In Procedure and Facilitation Points, “1. Read the following scenario: Ann's Bakery prepared several types of cakes to hand out at their grand opening event. Each bag contains pictures of the type of cakes that were served. There were several cookie cakes and mini cheesecakes for guests to sample. For each bag, you will open it, assemble the cakes, and use the pictures to answer the questions on your handout. 2. Students will work cooperatively to assemble the cakes in Bag 1. After assembling the cakes in Bag 1, they will do the following: a. Determine how many slices each cake is divided into and draw them on their Student Journal. b. Write the fraction for one slice of cookie cake. c. Write the improper fraction for the 17 slices of cake that were handed out. d. Find two possible combinations of slices that were handed out and write a number sentence that represents these possible combinations. e. Determine what their number sentences have in common. 3. Students will work cooperatively to assemble the cakes in Bag 2. After assembling the cakes for Bag 2, they will do the following: a. Determine how many slices each cheesecake is divided into and draw them on their Student Journal. b. Write the fraction for one slice of cheesecake. c. Develop two possible combinations of flavors if there are three slices of cheesecake and two flavors that were passed out. Write a number sentence to represent the combinations and label the fractions by flavor. d. While students are working cooperatively, use guided questions for struggling students. i. DOK-1 How do you know what fraction each cookie or cheesecake slice represents? ii. DOK-1 Do you have the same combination as everyone in your group? iii. DOK-1 What does “at least” mean? iv. DOK-1 Are these the only combinations? 4. After the Explore, invite the class to a Math Chat to share their observations and learning. Math Chat: DOK-1 How can we decompose an improper fraction? DOK-2 What did the fractions in your number sentences have in common? Explain why? DOK-2 Compare your combinations for the possible 17 slices that were handed out at the bakery from Bag 1. How do your combinations compare to another group’s combinations? Dok-2 Challenge them: Does this lesson have to be about circle-shaped cakes/pies? ” 

• Scope 7: Compare Fractions, Standards for Mathematical Practice and Explain, My Math Thoughts, Standard for Mathematical Practice, “MP.3 Construct viable arguments and critique the reasoning of others: Students will have the opportunity to make conjectures and justify their conclusions when determining the comparisons of two fractions. Counterexamples, such as thinking the sizes of the wholes being equal is not important, may be given when analyzing the reasoning and visual models of others.” My Math Thoughts, Student Handout: Question One: “Rudra has completed $\frac{2}{6}$ of her homework. Sedric has completed $\frac{1}{4}$ of the same homework assignment. Who has completed the most homework? Describe how you could answer the question by using benchmark fractions.” Question Two: “Use any fraction model to show the relationship between $\frac{2}{6}$, $\frac{1}{4}$, & $\frac{1}{2}$.” Question Three: “Corina has completed $\frac{4}{12}$ of her homework. She has completed exactly the same amount of homework as which other student? Use a model to justify your answer. Use the relationship of each of the fractions to to prove they are equivalent to $\frac{1}{2}$. There are many ways to model fractions. Why did you choose the fraction model you did for the previous question?”

• Scope 11: Problem Solve with Measurement, Evaluate, Decide and Defend, students construct viable arguments by creating a conjecture and perform an error analysis of provided student work. “Denise and Hank had to go grocery shopping for a school assignment and had to pick out fruits and vegetables that had a mass of less than a total of 2 kilograms. However, the scale at the store only weighed the fruits and vegetables in grams. Below is the data for the fruits and vegetables that each student picked out and their masses. What do the students need to do in order to find out if they correctly completed the assignment? Which student correctly completed the assignment?” Two tables are shown. One is labeled Denise and has 2 columns: Fruit and Vegetables & Mass. “Potato 280 g; Pineapple 1,166 g; Strawberries 385 g; Avocado 162 g.” The table labeled Hank has the same table headings with the data, “Peach 147 g; Banana 118 g; Cantaloupe 1,360 g; Grapes 632 g.” 

• Scope 13: Multiplication Models and Strategies, Standards for Mathematical Practice and Explore, Explore 2–Multiply Four-Digit by One-Digit Numbers: Area Models, Exit Ticket, Standards for Mathematical Practice, “MP.3 Construct viable arguments and critique the reasoning of others:Students are able to analyze a multiplication problem and make conjectures as to how to solve. They clearly and precisely justify their solution by providing evidence through visual representation or equations. Counterexamples may be given when analyzing the reasoning of others.”  On the Exit Ticket, “For each multiplication equation, draw an array to model the problem and use what you know about place value and multiplication to solve. Write equations to show your thinking. $6\times126=$ ___, $3,124\times5=$ ___”

The materials reviewed for STEMscopes Math Grade 4 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with the Math Practices across the year and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 10: Multiply Fractions by Whole Numbers, Explain, Show What You Know, Part 1, Student Handout, students build experience MP4 as they describe what they do with their models/strategies and how they relate to the problem. “Part 1: Multiplying Unit Fractions and Whole Numbers–Solve each problem:  Explain your reasoning using a model or strategy.  Write an equation with a variable to represent the problem.  Write a solution statement. Beth runs $\frac{2}{3}$ of a mile each day. What is the total distance that Beth runs in 5 days?” 

• Scope 15: Prime and Composite Numbers, Evaluate, Decide and Defend, Student Handout students build experience with MP4 as they draw models to represent the designs for a playground to help identify prime or composite numbers. “Granger Elementary School staff was told that they would be getting two new rectangular playground sets added to their recess field! One set would cover 15 square feet and the other set would cover 24 square feet. Luckily, the design of the playground set could be adjusted to best fit the playground needs of the students. The Playground Debate Do you agree with the following? The number 15 is prime and the 15-square-foot playground set can only have one rectangular design. The number 24 is composite and the 24-square-foot playground set can have more than one design. Use the space below to draw out possible playground designs for each set and explain your reasoning. However, it was discussed that the playground set that was 15 square feet could only have one design because 15 is an odd number, so it must be prime. The 24-square-foot playground set could have more than one design because 24 is an even number and must be composite. Use the space below to draw out possible playground designs for each set and explain your reasoning.”

• Scope 16: Multiplicative Comparisons in Multiplication and Division, Explore, Explore 3–Multiplication and Division Problem Solving, Exit Ticket, students build experience with MP4 as they model the situation with an appropriate representation and use an appropriate strategy. “The Final Purchase: Exit Ticket; Record your equation, comparison model, and solution statement for the purchase in the space below.” Problem: “You have been saving your money for a brand new bicycle. Sadly, the cost of the bike is still 6 times more than what you have saved up so far.” Image of a bike with a price tag of $72 is shown. “How much money do you have in savings?” Following there is a space for “Equation, Model, Solution Statement”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students use appropriate tools strategically as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 4: Addition and Subtraction Algorithms, Explore, Explore 3: Adding and Subtracting Strategies, Exit Ticket, students build experience with MP5 as they select strategies to solve word problems, and justify the most efficient strategy. “Read the scenarios below and solve them using the strategy that you think is the most efficient. Then justify the efficiency of your strategy and write a solution statement. A doughnut shop had 16,000 glazed doughnut holes. In a few hours, the doughnut shop sold 5,876 doughnut holes. How many doughnut holes are left at the shop? Equation: Your Strategy: Solution Statement: Justify the most efficient strategy:”

• Scope 11: Problem Solve with Measurement, Explore, Explore–Converting Units of Length, Standards for Mathematical Practice and Explain, Show What You Know Part 1, Print Files, Standards for Mathematical Practice: “Students use measurement tools for mass, liquid volume, and distance to understand the relative size of units within a system and to express measurements given in larger units in terms of smaller units.” Show What You Know Part 1, “The 4th-grade classes participated in the Measurement Olympics. Each station had a different activity and featured various measuring tools. Read the results on each recording card and convert each measurement to determine the winner.” A chart with three columns is shown. The first column has the names of students and the next two columns are the length and total inches and only one measurement is given and students must find the other. The following data is shown: “Andrew, Total inches: 49 in.; Sylvester, Length: 3 ft. 11 in.;  Olivia, Length: 3 ft. 2 in.; Maria, Length: 3 ft. 7 in.; Oscar, Total: 40 in. ___ jumped the farthest and is the winner.” 

• Scope 19: Angles: Explore, Explore1–Angles as Fractions of a Circle, Student Journal, students build experience with using a protractor accurately to create angles. “Part I: The 360, Use the circle below to record the pieces of your paper plate. Record the measure of each angle in degrees.” Image of a circle with a point in the center indicated. “Part II: Dance Moves Draw and label your dance design.” Image of a circle with a point in the center indicated.

The materials reviewed for STEMscopes Math Grade 4 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with the Math Practices across the year and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. MP6 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students attend to precision as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 4: Addition and Subtraction Algorithms, Evaluate, Skills Quiz, Problems 1-5, students attend to precision as they calculate equations accurately. "$6,037+4,510=$ ___; $3,006+2,872=$ ___; $8,512+4,284=$ ___;  $6,199+4,832=$ ___; $4,622+1,307=$ ___”

• Scope 12: Represent Measurement with Line Plots, Explore, Explore 2–Problem Solving Using Line Plots, Student Journal, student build experience with MP6 as they use clear and precise language to specify measurements. Students will show their work for each of the Station Cards. “Client A - Insect Length How long was the longest insect? ___ How long was the shortest insect? ___ What is the difference between the longest and shortest insect? ___ How many insects were measured in this study?___.” 

• Scope 20: Points, Lines and Angles, Explore, Explore 1–Draw and Identify Points, Lines, Rays and Angles, Print Files, Student Journal, engages students in MP6 as they accurately draw and describe objects. “Constellation Attributes: Use the materials provided by your teacher to create each attribute. Draw a model of each attribute in the table below. Then write a short description of your model.” This is followed by a table with columns labeled “Attribute, Model, Description, Name It” and the rows labeled: “Point, Ray, Line, Line segment, Angle” and then, “In the space below, draw an object found in the classroom or school that has one of the following attributes. Use letters to label the points in your drawing. Be sure to name each attribute”. Below this is a chart with columns labeled: “Point, Ray, Line, Line Segment, Angle” and blanks for students to name it.

The materials reviewed for STEMscopes Math Grade 4 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with the Math Practices across the year and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and make use of structure as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 2: Place Value of Whole Numbers, Explore, Explore 2–Reading and Writing Multi-Digit Whole Numbers, Procedure and Facilitation Points, students build experience with MP7 as they relate the structure of the base-ten system to recognize that in a multi-digit whole number, a digit in the one place represents ten times what it represents in the place to its right. “... 3. Introduce the scenario: BREAKING NEWS! There has been a discovery of a massive variety of precious gemstones found underground–the biggest finding in history! You are a media manager, and you can’t wait to report this new discovery to the world. Different media outlets come with their own forms of communication, though. You will either receive the number of each place value or the amount in standard form. However, the websites and news reporters take in information in different forms. The news website uses a coding system, so it will need the number converted to expanded form to input the values into the site. The news reporter reads from a teleprompter, so they will need the number in word form to read it in the news report. Your challenge is to convert the information you’re given into a form that can be used by other news outlets so you can share in the excitement of this discovery and update the world! 4. Students will read the gemstone type and amount on each Gemstone Discovery Card and work with their group to build the number on the Place Value Chart with Placed Value Disks. 5. Encourage students to label the place values on their Place Value Charts to help distinguish in what place value the digits belong. They should use the chart as a work mat to help guide their thinking. a. Ask guiding questions such as the following: i. DOK-2 How does building the number with disks help you convert the number into a certain form? ii. DOk-1 If there is zero of a place value, do we have to write that into expanded form? 6. Students should take one card from the bag, build it on their Place Value Charts, then convert it into standard form, expanded form, and word form on their Student Journal. 7.” 

• Scope 5: Represent and Compare Decimals, Explore, Explore 1–Decimals Notation for Denominators of 10, Procedure and Facilitation Points, students build experience with MP7 as they look for patterns in the base-ten number system, noting that the value of each place-value position is 10 times the value of the place to its right and one-tenth of the value of the place to its left. “... 2. Begin the lesson by reviewing the relationships between base ten blocks. a. Hold up a flat, a rod, and a small unit cube. Ask the following question: i. DOK-1 If the flat is considered one whole, what are the values of the other pieces? 3. Pass out a bundle of 100 craft sticks to each group along with a few base ten block flats and rods. Explain that students will be assembling mini sculptures that can be painted and used to decorate bedrooms, playrooms, offices, and more! 4. Show students how to build one sculpture. Each sculpture should be built from 10 craft sticks and be arranged as shown below, with five layers of two sticks. 5. Set the timer for 30 seconds. Instruct student groups to build as many sculptures as they can in 30 seconds. Tell them the whole group must work on one sculpture at a time. 6. When the timer goes off, have students stop building their sculptures. 7. Have students use the base ten blocks to build a model of how many sculptures they were able to build using the flat as one whole.”

• Scope 21: Properties of Two-Dimensional Figures, Explore, Explore 2–Identifying Types of Triangles, Procedure and Facilitation Points, students build experience with MP7 as they notice and describe the common attributes within categories and subcategories of shapes. “1. Introduce students to the scenario: a. The local art museum is hosting a contest for its latest exhibit! It wants to display an art piece that shows the uniqueness of triangles. It has asked that the art piece display how angles can be used to classify types of triangles. 2. Briefly review types of angles students learned in the previous scope. Ask the following questions: a. DOK-1 What are the three different types of angles? b. DOK-1 What is an acute angle? c. DOK-1 What is a right angle? d. DOK-1 What is an obtuse angle? 3. Give each student a copy of the Student Journal and a note card. Give each group a set of triangles, a pair of scissors, and a sheet of construction paper. Ask the following question: a. DOK-1 How could we use a note card to help us identify angles? 4. If needed, model for students how to use the note card to identify each angle. 5. Review how to name angles using symbols (∠ABC). Remind students that the middle letter (point) is the vertex of the angle being referred to. Ensure students understand that there are two ways to name an angle. The first and last letter of the angle name can be switched and still be referring to the same angle if the middle letter stays the same. 6. Have students fold their sheet of construction paper into three sections (like a brochure). Have students title the sections as follows: a. All Acute Angles b. One Right Angle c. One Obtuse Angle 7. Have students cut out the triangles from each section on their Student Journal page as well as one additional triangle that fits the criteria. 8. Have students draw at least two triangles from each section on their Student Journal page as well as one additional triangle that fits the criteria.”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and express regularity in repeated reasoning as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 3: Rounding, Explore, Explore 2–Rounding Using Reasoning, Procedure and Facilitation Points, students build experience with MP8 as they use repeated reasoning to make generalizations about rules for rounding numbers. “1. Present students with the following scenario: Your family loves to travel all over! You like to keep a journal estimating the number of miles your family travels each month. You also want to know the difference between the lengths of the trip you took each month. You want to get as close to the exact number of miles quickly so you choose to quickly round the numbers. So you use your reasoning skills to choose which place value it makes more sense to round to given the distances to each destination. 2. DOK-2 Discuss with students: What can we consider when choosing which place value to round to? … 6. Challenge students to try and do the calculations mentally. 7. Tell students they don’t need to find the exact amount. It is okay to calculate the total by strategically choosing numbers that are close. 8. Allow students to share their strategy and solution with their group members. Students should then record their mental calculation strategy on their Student Journal. 9. Encourage students to repeat the same process and try to mentally calculate the difference between the distances traveled. 10. Students should share their strategy and solution with their groups and record them on their Road Trip! sheet. 11. Students should rotate to the next month and repeat the same process on your cue.”

• Scope 9: Add and Subtract Fractions and Mixed Numbers, Explore, Explore 2–Add Fractions and Mixed Numbers with Like Denominators, Procedure and Facilitation Points, students build experience with MP8 as they use a variety of visual fraction models to explain calculations and make generalizations about sums and differences of fractions. “Part I … 3. Distribute a set of Training Cards and three sets of fraction circles or three sets of fraction tiles to each group. 4. Present the students with the following scenario: Some fellow runners are training to run races. Each day they run a different amount depending on their training schedule. It is our job to figure out how many miles each runner has completed for their training. 5. Have students look at the Megan training card. Read the scenario together. 6. Instruct the students to work together with their group and use their fraction tiles or circles to solve the scenario. … 11. Next, read the Calvin scenario with the class and encourage the groups to work together to solve it. 12. Allow students to discuss the strategies they used to find the total distance Calvin ran. 13. Have the students record the model, addition sentence, and solution statement that they used to solve for Calvin in their Student Journal. Part II … 3. Present the following scenario. There are different kinds of racing competitions all over the world and in our communities. In these different races, there are different amounts of laps or miles that the competitors complete. It is your job to help these competitors figure out how many laps or miles they completed. … 6. As the students work at each station, monitor students for understanding by asking some of the following questions. a. DOK-2 How can we represent the distance given in the scenario? b. DOK-2 How do we represent the joining of the distances completed? c. DOK-2 How can we use the model to help us solve the scenario?” 

• Scope 22: Generate Patterns, Explore, Explore 2–Number Patterns, Procedure and Facilitation Points, students build experience with MP8 as they examine patterns and relate the pattern to a rule. “1. Discuss what students already know about patterns. a. DOK-1 What is a pattern? b. DOK-2 Where do you see patterns? 2. Read the following scenario. Your cousins, Amari and Flynn, share the same birthday. Your family would like to throw a celebration extravaganza for both of them! You need to help your family plan the party! 3. Distribute the Student Journal and show students the chosen manipulatives. Explain that they will use manipulatives to model the relationships described at each station. 4. Students will use their model to identify the rule and continue the pattern. 5. Ask the following questions. a. DOK-2 How do you think you can find out what is happening with the numbers in the right column? 6. Have each group start at a different station. Monitor and check for understanding as students are working. Ask the following guiding questions. a. DOK-1 How did you know what information to put in the columns? b. DOK-1 What did you notice about the number pattern in this scenario? c. DOK-1 how were you able to figure out the situations that weren’t on the table?”

The materials reviewed for STEMScopes Math Grade 4 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Within each Scope, there is a Home dropdown menu, where the teacher will find several sections for guidance about the Scope. Under this menu, the Scope Overview has the teacher guide which leads the teacher through the Scope’s fundamental activities while providing facilitation tips, guidance, reminders, and a place to record notes on the various elements within the Scope. Content Support includes Background Knowledge; Misconceptions and Obstacles, which identifies potential student misunderstandings; Current Scope, listing the main points of the lesson, as well as the terms to know. There is also a section that gives examples of the problems that the students will see in this Scope, and the last section is the Coming Attractions which will describe what the students will be doing in the next grade level. Content Unwrapped provides teacher guidance for developing the lesson, dissecting the standards, including verbs that the students should be doing and nouns that the students should know, as well as information on vertical alignment. Also with each Explore, there is a Preparation list for the teacher with instructions for preparing the lesson and Procedure and Facilitation Points which lists step-by-step guidance for the lesson. Examples include:

• Scope 3: Rounding, Explore, Explore 1–Money in the Bank!, Procedure and Facilitation Points, provides teacher guidance. “Part I 1. Have students gather around the number line with multiples of 1,000. Ask the following question: DOK-1 What multiples does this number line show? 2. Read the following scenario: You and your group have started a toy business. The number on which your beanbag lands is the amount you will report as your estimated savings each week from the sales of your toys. You need an estimate of your business’s profit - how much money you are able to save each week–in order to be able to buy more materials to make new toys. 3. Ask the following questions: DOK-2 When we want an estimate, what can we do? DOK-2 When is rounding numbers useful? 4. Choose one student from each group to demonstrate how to toss the beanbag onto the 1,000 number line.” 

• Scope 10: Multiply Fractions by Whole Numbers, Home, Scope Overview, Teacher Guide, Engage Activities, Assessing Prior Knowledge. Teachers, “Before beginning the lesson, students’ prior knowledge is assessed. The teacher assesses using visual models to represent multiplication. Students will choose a visual model such as an area model, array, or bar model to solve a real-world problem.” 

• Scope 17: Problem Solve Using the Four Operations, Explore, Explore 4–Problem Solve Using the Four Operations (Level 3), Procedure and Facilitation Points. Teachers follow these steps:  “1. Distribute a resealable bag of Scenario Cards to each pair. 2. Distribute a Student Journal, a whiteboard, and a dry erase marker to each student. 3. Tell students that for every problem, they will need to draw a strip diagram on their whiteboards, build an equation using a letter for an unknown quantity (maybe even two letters for two unknown quantities!), estimate the solution, and then solve. 4. Remind students to take their time and do one step at a time.”

The materials reviewed for STEMScopes Math Grade 4 meet expectations for containing adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

Each Scope has a Content Overview with a Teacher Guide. Within the Teacher Guide, information is given about the current Scope and its skills and concepts. Additionally, each Scope has a Content Support which includes sections entitled: Misconceptions and Obstacles, Current Scope, and Coming Attractions. These resources provide explanations and guidance for teachers. Examples include:

• Scope 3: Rounding, Home, Content Overview, Teacher Guide, Scope Summary. It states, “In this Scope, students learn how to use place value understanding to round multi-digit whole numbers to any place collaboratively with real-world scenarios.”

• Scope 8: Equivalent Fractions, Home, Content Support, Current Scope. It states, “Students will use visual models such as area models and number lines as they generate and explain equivalent fractions. Fractions include denominators of 2, 3, 4, 5, 6, 8, 10, and 100. Students will express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100. This is used in a problem-solving context.”

• Scope 14: Division Models and Strategies, Home, Content Support, Coming Attractions. It states, “Grade five extends division to two-digit divisors and includes dividends with decimals to hundredths. They continue using concrete models or drawings based on place value and the properties of operations. Students relate a division strategy to a written method, consistently checking the reasonableness of their solutions. Sixth grade applies and extends students’ understanding of division to divide fractions by fractions. They fluently divide multi-digit numbers and decimals using the standard algorithm.”

• Scope 22: Generating Patterns, Content Overview, Future Expectations. It states,  “In fifth grade, students generate two numerical patterns using two given rules, and they identify relationships between the corresponding terms. Fifth-grade students form ordered pairs of the corresponding terms and graph them on a coordinate plane.”

The materials reviewed for STEMScopes Math Grade 4 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Correlation information is present for the mathematics standards addressed throughout the grade level and can be found in several places including a drop-down Standards link on the main home page, within teacher resources, and within each Scope. Explanations of the role and progressions of the grade-level mathematics are present. Examples include:

• In each Scope, the Scope Overview, Scope Content, and Content Unwrapped provides opportunities for teachers to view content correlation in regards to the standards for the grade level as well as the math practices practiced within the Scope. The Scope Overview has a section entitled Student Expectations listing the standards covered in the Scope. It also provides a Scope Summary. In the Scope Content, the standards are listed at the beginning. This section also identifies math practices covered within the Scope. Misconceptions and Obstacles, Current Scope, and Background Knowledge make connections between the work done by students within the Scope as well as strategies and concepts covered within the Scope. Content Unwrapped again identifies the standards covered in the Scope as well as a section entitled, Dissecting the Standard. This section provides ideas of what the students are doing in the Scope as well as the important words they need to know to be successful.

• Teacher Toolbox, Essentials, Vertical Alignment Charts, Vertical Alignment Chart Grade K-5, provides the following information: “How are the Standards organized? Standards that are vertically aligned show what students learn one grade level to prepare them for the next level. The standards in grades K-5 are organized around six domains. A domain is a larger group of related standards spanning multiple grade levels shown in the colored strip below: Counting and Cardinality, Operations and Algebraic Thinking,  Number and Operations in Base Ten, Number and Operations–Fractions, Measurement and Data, Geometry.” Tables are provided showing the vertical alignment of standards across grade levels.

• Scope 3: Rounding, Home, Content Unwrapped, Implications for Instruction, states that “Learners often confuse estimation with guessing. Estimation involves reasoning and thought. When possible, include real-world situations in which estimation would be used. Prior to this grade level, students have rounded to the nearest 10 or 100 to estimate solutions to addition and subtraction problems.Students often try to memorize “rounding rules” to know when to round. However, this often does not provide them with a purpose for rounding. Rounding should be presented in contextual situations with an emphasis on the reasoning skills associated with it. It would be beneficial to identify conceptual misunderstandings of place value.Students will be rounding numbers up to 1,000,000 to any place value. Typically, they have an easier time rounding to place values in the outermost left and often struggle when rounding to place values in the middle of a number.”

• Scope 15: Prime and Composite, Home, Key Concepts, Key Concepts, states “I can demonstrate an understanding of the relationship between factors and multiples of whole numbers. I can distinguish between a prime number and a composite number. I can use a variety of strategies and tools (build rectangular arrays and area models, use a hundreds chart, use counters) to find all the factor pairs of a number in the range 1–100, and I can use the number of factor pairs to determine if the number is prime or composite. I can notice patterns within the multiples of numbers on a hundreds chart. I can generate multiples by skip counting by each factor, and I can keep track of the number of factors within the number. I can recognize that a whole number is a multiple of each of its factors.”

The materials reviewed for STEMscopes Math Grade 4 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

The Teacher Toolbox contains an Elementary STEMscopes Math Philosophy document that provides relevant research as it relates to components for the program. Examples include:

• Teacher Toolbox, Essentials, STEMscopes Math Philosophy, Elementary, Learning within Real-World, Relevant Context, Research Summaries and Excerpts, states, “One of the major issues within mathematics classrooms is the disconnect between performing procedural skills and knowing when to use them in everyday situations. Students should develop a deeper understanding of the mathematics in order to reason through a situation, collect the necessary information, and use the mechanics of math to develop a reasonable answer. Providing multiple experiences within real-world contexts can help students see when certain skills are useful. “If the problem context makes sense to students and they know what they might do to start on a solution, they will be able to engage in problem solving.” (Carpenter, Fennema, Loef Franke, Levi, and Empson, 2015).

• Teacher Toolbox, Essentials, STEMscopes Math Philosophy, Elementary, CRA Approach, Research Summaries and Excerpts, states, “CRA stands for Concrete–Representational–Abstract. When first learning a new skill, students should use carefully selected concrete materials to develop their understanding of the new concept or skill. As students gain understanding with the physical models, they start to draw a variety of pictorial representations that mirror their work with the concrete objects. Students are then taught to translate these models into abstract representations using symbols and algorithms. “The overarching purpose of the CRA instructional approach is to ensure students develop a tangible understanding of the math concepts/skills they learn.” (Special Connections, 2005) “Using their concrete level of understanding of mathematics concepts and skills, students are able to later use this foundation and add/link their conceptual understanding to abstract problems and learning. Having students go through these three steps provides students with a deeper understanding of mathematical concepts and ideas and provides an excellent foundational strategy for problem solving in other areas in the future.” (Special Connections, 2005).” STEMscopes Math Elements states,  “As students progress through the Explore activities, they will transition from hands-on experiences with concrete objects to representational, pictorial models, and ultimately arrive at symbolic representations, using only numbers, notations, and mathematical symbols. If students begin to struggle after transitioning to pictorial or abstract, more hands-on experience with concrete objects is included in the Small Group Intervention activities.”

• Teacher Toolbox, Essentials, STEMscopes Math Philosophy, Elementary, Collaborative Exploration, Research Summaries and Excerpts, states, “Our curriculum allows students to work together and learn from each other, with the teacher as the facilitator of their learning. As students work together, they begin to reason mathematically as they discuss their ideas and debate about what will or will not work to solve a problem. Listening to the thinking and reasoning of others allows students to see multiple ways a problem can be solved. In order for students to communicate their own ideas, they must be able to reflect on their knowledge and learn how to communicate this knowledge. Working collaboratively is more reflective of the real-world situations that students will experience outside of school. Incorporate communication into mathematics instruction to help students organize and consolidate their thinking, communicate coherently and clearly, analyze and evaluate the thinking and strategies of others, and use the language of mathematics.” (NCTM, 2000)

• Teacher Toolbox, Essentials, STEMscopes Math Philosophy, Elementary, Promoting Equity, Research Summaries and Excerpts, states,  “Teachers are encouraged throughout our curriculum to allow students to work together as they make sense of mathematics concepts. Allowing groups of students to work together to solve real-world tasks creates a sense of community and sets a common goal for learning for all students. Curriculum tasks are accessible to students of all ability levels, while giving all students opportunities to explore more complex mathematics. They remove the polar separation of being a math person or not, and give opportunities for all students to engage in math and make sense of it. “Teachers can build equity within the classroom community by employing complex instruction, which uses the following practices (Boaler and Staples, 2008): Modifying expectations of success/failure through the use of tasks requiring different abilities, Assigning group roles so students are responsible for each other and contribute equally to tasks, Using group assessments to encourage students' responsibility for each other's learning and appreciation of diversity” “A clear way of improving achievement and promoting equity is to broaden the number of students who are given high-level opportunities.” (Boaler, 2016) “All students should have the opportunity to receive high-quality mathematics instruction, learn challenging grade-level content, and receive the support necessary to be successful. Much of what has been typically referred to as the "achievement gap" in mathematics is a function of differential instructional opportunities.” (NCTM, 2012).” STEMscopes Math Elements states, “Implementing STEMscopes Math in the classroom provides access to high quality, challenging learning opportunities for every student. The activities within the program are scaffolded and differentiated so that all students find the content accessible and challenging. The emphasis on collaborative learning within the STEMscopes program promotes a sense of community in the classroom where students can learn from each other.”

The materials reviewed for STEMScopes Math Grade 4 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Teacher Toolbox provides an Elementary Materials List that provides a spreadsheet with tabs for each grade level, K-5. Each tab lists the materials needed for each activity. Within each Scope, the Home Tab also provides a material list for all activities. It allows the teacher to input the number of students, groups, and stations, and then calculates how many of each item is needed. Finally, each activity within a Scope has a list of any materials that are needed for that activity. Examples include:

• Scope 5: Represent and Compare Decimals, Elaborate, Fluency Builder–Who is the Greatest?, Materials, “Printed, 1 Instruction Sheet (per pair), 1 Set of Digit Cards (per pair), 1 Work Mat and Recording Sheet (per student), Consumable, 1 Bag or envelope (per group)”

• Scope 10: Multiply Fractions by Whole Numbers, Explore, Explore 2–Multiplying Multiples of Unit Fractions, Materials, “Printed, 1 Student Journal (per student), 1 Exit Ticket (per student), 2 Sets of Station Cards (per class), Reusable, 18 Sets of fraction tiles (per class), 10 Sets of fraction circles (per class), 8 Rulers (per class),Consumable, 2 Strips of manila paper, measuring 3 x 18 inches (per student), 2 Dispensers with clear tape (per class)”

• Scope 15: Prime and Composite Numbers, Explore, Explore1–Finding Factor Pairs, Materials, “Printed, 1 Student Journal (per student), 1 Set of Scenario Cards (per group), 1 Exit Ticket (per student), Reusable, 100 Color tiles (per station)”

The materials reviewed for STEMScopes Math Grade 4 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

In Grade 4, each Scope has an activity called Decide and Defend, an assessment that requires students to show their mathematical reasoning and provide evidence to support their claim. A rubric is provided to score Understanding, Computation, and Reasoning. Answer keys are provided for all assessments including Skills Quizzes and Technology-Enhanced Questions. Standards-Based Assessment answer keys provide answers, potential student responses to short answer questions, and identifies the Depth Of Knowledge (DOK) for each question. 

After students complete assessments, the teacher can utilize the Intervention Tab to review concepts presented within the Scopes’ Explore lessons. There are Small-Group Intervention activities that the teacher can use with small groups or all students. Within the Intervention, the lesson is broken into parts that coincide with the number of Explores within the Scope. The teacher can provide targeted instruction in areas where students, or the class, need additional practice. The program also provides a document in the Teacher Guide for each Scope to help group students based on their understanding of the concepts covered in the Scope. The teacher can use this visual aide to make sure to meet the needs of each student. Examples include:

• Scope 3: Rounding, Evaluate, Standards-Based Assessment, Answer Key, Question 2, provides a possible way a student might complete the problem. “What is 323 rounded to the nearest ten? Use the number line to explain your thinking. (DOK-3) 323 rounded to the nearest ten is 320, because it is closer to that value than it is to 330.”  (4.NBT.3)

• Scope 12: Represent Measurements with Line Plots, Evaluate, Standards-Based Assessment, Answer Key, Question 8, Part B, provides a possible solution a student might provide. “How many textbooks weigh less than one pound? Write your answer in the box. Use equations or text to justify your answer. (DOK-3), 7 textbooks Sample reasoning: There are two dots above one-fourth, one dot above three-eights, three dots above one-half, and one dot above three-fourths. $2+1+3+1=7$.” (4.MD.4)

• Scope 17: Problem Solving Using the Four Operations, Intervention, Small-Group Intervention states, “Part I: Addition and Subtraction, 1. Read the following word problem to the group: a. Jasmine has a goal to read 345 pages in a chapter book over a four-day period. She reads 50 pages on Monday, 66 pages on Tuesday, and 140 pages on Wednesday. How many pages will Jasmine need to read on Thursday to reach her goal of 355 pages? 2. Allow time for students to discuss the word problem in their pairs. 3. Ask the following questions: a. What is the question asking us to find? It’s asking how many pages Jasmine needs to read on Thursday to reach her goal of 345 pages. B. What information is needed to solve the problem? She read 50 pages on Monday, 66 pages on Tuesday, and 140 pages on Wednesday. 4. Distribute the whiteboard materials to the students and prompt them to draw a long rectangle on their board…”

The materials reviewed for STEMscopes Math Grade 4 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. 

Assessment opportunities are included in the Exit Tickets, Show What You Know, Skills Quiz, Technology-Enhanced Questions, Standards-Based Assessment, and Decide and Defend situations. Assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types, including multiple choice, multiple response, and short answer. While the MPs are not identified within the assessments, MPs are described within the Explore sections in relation to the Scope. Examples include:

• Scope 5: Represent and Compare Decimals, Evaluate, Skills Quiz, Question 3 and 6, provides opportunities for students to demonstrate the full intent of MP7, “Look for and make use of structure, as they look for patterns when converting decimals to fractions noting the tenths and hundredths.” “Convert each decimal to a fraction. 3. 0.1 = ___, 6. 0.85 = ___”

• Scope 16: Multiplicative Comparisons in Multiplication and Division, Evaluate, Standards- Based Assessment, Print Files, Student Handout, provides opportunities for students to demonstrate the full intent of the standard, 4.OA.2, “Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.” “Read each question. Then follow the directions to answer each question. Mark your answers by circling the correct answer choices. If a question asks you to show or explain your work, you must do so to receive full credit.” Question one: “1. Atticus and Nina kept track of how many text messages they each received in one day. Atticus received 20 text messages, while Nina received 5. Which two statements are correct? A. Atticus received 4 times as many messages as Nina. B. Atticus received 15 times as many messages as Nina. C. Atticus received 15 more messages than Nina. D. Nina received one-fifth of the number of messages as Atticus.” Question 2: “ Scott has 2 surfboards. His son has 3 more than he has. Which equation represents the number of surfboards Scott’s son has? A. $2\times3=n$ B. $n+3=2$ C. $n+2=3$ D. $2+3=n$”. Question 5, “A multiplicative comparison is shown below.” A bar model with 4 equal size boxes with 7 labeled in each one and a second bar below that with one 7 labeled in the box. “Which three statements describe this comparison? A. 28 is 4 times as much as 7. B. There are 5 sevens in 28. C. 7 is one-fourth of the value of 28. D. $7\times4=28$”.

• Scope 19: Angles, Explore, Explore 1, Print Files, Exit ticket, provides opportunities for students to demonstrate the full intent of the standard 4.MD.5a, An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through $\frac{1}{360}$ of a circle is called a "one-degree angle," and can be used to measure angles. “Design your best dance moves by using at least one whole-circle turn, one straight-angle turn, and one right-angle turn! Draw each angle and label it with its name, degree measure, and fraction of a circle.” Below the directions are 6 boxes connected with arrows for students to complete.

The materials reviewed for STEMscopes Grade 4 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Within the Teacher Toolbox, under Interventions, materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. Within each Explore section of the Scopes there are Instructional Supports and Language Acquisition Strategy suggestions specific to the Explore activity. Additionally, each Scope has an Intervention tab that provides support specific to the Scope. Examples include:

• Teacher Toolbox, Interventions, Interventions–Adaptive Development, Generalizes Information between Situations, supplies teachers with teaching strategies to support students with difficulty generalizing information. “Unable to Generalize: Alike and different–Ask students to make a list of similarities and differences between two concrete objects. Move to abstract ideas once students have mastered this process. Analogies–Play analogy games related to the Scope with students. This will help create relationships between words and their application. Different setting - Call attention to vocabulary or concepts that are seen in various settings. For example, highlight vocabulary used in a math problem. Ask students why that word was used in that setting. Multiple modalities– Present concepts in a variety of ways to provide more opportunities for processing. Include a visual or hands-on component with any verbal information.”

• Scope 5: Represent and Compare Decimals, Explore, Explore 3–Represent and Compare Decimals, Instructional Supports states, “1. If students are struggling with the process of comparing decimals, remind students that the instructions ask them to cover each place value with a sheet of paper, working from left to right. It may be necessary to provide examples with smaller numbers, starting with whole-number examples first and working toward the tenths place, to ensure they understand the process. Ensure students understand that the decimal points must always be aligned vertically so that each place value is in line properly. 2. If students are struggling with writing a comparison statement using symbols, revisit the prerequisite skill of comparing only one decimal. Then have them try comparing two decimals. 3. If students are struggling with the greater-than and less-than symbols, it might be helpful to present the concept in multiple ways. First, draw pictures of two groups of objects on the board with space between them. Have three sticky notes available on which you’ve drawn <, >, and =. Ask students which group has more (greater than), and put the appropriate sticky note between the groups. Repeat for less than and equal to. Explain that the symbol always opens to the number that is greater. You could also make a game by writing two numbers on the board and asking students to write the correct symbol on their personal whiteboards and hold them up. 4. If a student is struggling with creating his or her own number line, provide the student with a premade number line with benchmark numbers labeled.”

• Scope 19: Angles,  Explore, Explore 4–Additive Angles, Instructions Supports states, “1. If students are struggling to understand the concept of adjacent angles, it might be helpful to demonstrate using the cut paper plates from Explore 1. 2. Some students may benefit from using the 90-degree and 180-degree paper plate cutouts when determining angles. The frame decorations could be laid on top of the paper plate pieces. 3. Remind students that it might be easier to rotate their cutouts while they measure the angles, rather than try to read a protractor that’s at an awkward angle. 4. For Part II, some students may find it beneficial to use base ten blocks or other manipulatives to help them add and subtract the angle measurements.”

The materials reviewed for STEMscopes Math Grade 4 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Within each Scope, Scope Overview, Teacher Guide, a STEMscopes Tip is provided. It states,  “The acceleration section of each Scope, located along the Scope menu, provides resources for students who have mastered the concepts from the Scope to extend their mathematical knowledge. The Acceleration section offers real-world activities to help students further explore concepts, reinforce their learning, and demonstrate math concepts creatively.” Examples include:

• Scope 9: Add and Subtract Fractions and Mixed Numbers, Acceleration, Math Today–Winbot, Question 3 states,  “A company bought a Winbot for their office building. The company has two buildings that have eight windows each that need window cleaning. On Day 1 the Winbot cleaned $1\frac{3}{8}$ of the windows on the buildings. How much of the windows does the Winbot still need to clean?”

• Scope 15: Prime and Composite Numbers, Acceleration, Math Today–Melting Away,  Question 1 states, “One of the lakes the melting glaciers feeds into in Peru is Lake Palcacocha. The volume of this lake is 34 times what it was in 1947. Is 34 a prime or composite number? Explain your reasoning.”  Question 2, “Rising water levels in places such as Lake Palcacocha have increased the size of lakes and decreased the amount of land. The lake covers the land at a rate of 25 square feet per year. Is 25 a prime or composite number? Explain your reasoning.” 

• Scope 21: Properties of Two-Dimensional Figures, Acceleration, Math Today–Washington Monument, Question 1 states, “What type of angle is formed by two adjacent blades of the helicopter?” Question 2, Can a line of symmetry be drawn in the shape that makes up the top section of the monument? If so, draw the shape and the line of symmetry.” Question 3, “The tall, non triangular shape that makes up most of the monument has vertical edges. Are these edges parallel? Explain your answer?” Question 4, “A structure is being built to honor you! Draw and label a blueprint of your structure below. You must include one quadrilateral with at least one set of parallel sides, one acute angle, and one obtuse angle. Your structure must have symmetry. Your structure must also include a right triangle.”

The materials reviewed for STEMscopes Math Grade 4 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics. 

Within the Teacher Toolbox, the program provides resources to assist MLLs when using the materials. The materials state, “In the curriculum, we have integrated resources to support teachers and families. Below are a few features and elements that can be used to support students at their level and provide an opportunity for families and caregivers to engage in student learning.” Examples include but are not limited to:

• “Proficiency Levels by Domain – In this section, you will find a snapshot of language application across domains at different proficiency levels. Teachers can use this tool to help identify a student’s English proficiency level by analyzing how students are able to interpret and produce language.”  

• “Working on Words – This open-ended activity allows students to take agency and accountability for their growing vocabulary. This activity also encourages making relevant, personal connections to new terms in different ways, such as identifying cognates.” 

• “Sentence Stems/Frames – Students are able to practice engaging in purposeful discussion. These sentence stems and sentence frames can be used for different intents, such as asking for clarification, defending their thinking, and explaining their responses.” 

• “Integrated Accessibility Features – Across the curriculum, we have embedded tools that allow students to listen to text being read, find the definition of words in the moment, make notes, and highlight words and phrases.” 

• “Parent Letters – Each scope includes a letter tailored to caregivers in which the content of a scope, including its vocabulary, is explained in simplified terms. Within the Parent Letters, we have included an activities section called Tic-Tac-Toe–Try This at Home that students can engage in along with their families. This letter is written in two languages.” 

• “Tiered Supports – Within each Explore lesson, we have included tiered supports and strategies that can be applied during the lesson for students at each proficiency level. These range in focus across all domains.” 

• “Language Connections – Every scope has three Language Connection activities, one at each proficiency level. Language Connections meets the students at their proficiency level by providing teachers with prompts to support students in demonstrating their understanding in each language domain.” 

• “Virtual Manipulatives – Students are able to use these across the curriculum to help them justify their answers when expressive language may be limited. These can also be used as tools for creating meaningful connections to vocabulary terms and skills.” 

• “Visual Glossary/Picture Vocabulary – Students are able to combine visual representations and mathematical terms using student-friendly language.” 

• “Distance Learning Videos – Major skills and concepts are broken down in these student- facing videos. Students and caregivers alike can engage in the activities at home at their own pace and incorporate familiar objects. In this way, students can apply their own language to math.”

• “My Math Thoughts/Math Story – These literary elements give students the opportunity to practice reading and writing about math. Students can apply reading strategies to aid with comprehension and practice not just math vocabulary, but situational vocabulary as well.”

Guidance is also provided throughout the scopes to guide the teacher. Examples include:

• Scope 4: Addition and Subtraction Algorithms, Explore, Explore 1–Multi-Digit Addition, Language Acquisition Strategy provides support for students who read, write, speak a different language than English to engage in the content. “The following Language Acquisition Strategy is supported in this Explore activity. See below for ways to support a student's English language development. Students apply a variety of different learning strategies, such as asking for assistance, interpreting nonverbal cues, and using context clues to derive and describe the meaning of unknown words.During discussion, check for understanding of the following words: modeling, strategy, algorithm, sum, and difference. Challenge students to come up with their own definitions and descriptions of these terms in their groups.Teach students the self-advocacy skills of requesting assistance, learning how to interpret body language or facial expressions that aren't necessarily spoken, and having students repeat back to you or each other what they understand a word or phrase to mean. This allows you to check for understanding. It also gives the students an opportunity to use the new words in sentences and receive specific feedback.”

• Scope 6: Compose and Decompose Fractions and Mixed Numbers, Explore, Explore 1–Joining and Separating Parts of a Whole, Language Acquisition Strategy provides support for students who read, write, speak a different language than English to engage in the content. “The following Language Acquisition Strategy is supported in this Explore activity. See below for ways to support a student's English language development. Students learn to spell English words correctly using patterns and English rules with continued accuracy and improvement. After the lesson, have students create a word cloud of the most commonly used terms in the lesson. The more they used the word in their answers, the bigger the word will be in the cloud. Say the words fourths, sixths, eighths, etc. to the class. Let the students listen to the sounds in the word. Have the students practice speaking the words with a partner and take turns listening and speaking.”

• Scope 15: Prime and Composite Numbers, Explain, Picture Vocabulary, Flash Cards, Student Handout provides support for students who read, write, speak a different language than English to engage in the content. The student handout provides models and explanations in terms such as factors and multiples for students to understand. This resource is also available in Spanish.

The materials reviewed for STEMscopes Math Grade 4 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods. Examples include:

• Scope 5: Represent and Compare Decimals, Explore, Virtual Manipulatives–Base Ten Blocks  provides virtual base ten block manipulatives to support students as they represent and compare decimals. 

• Scope 8: Division Models and Strategies, Explore, Skills Basics–Ways to Represent Division, Procedure and Facilitation Points provides for students’ active participation in content through the use of manipulatives. “Give one Strategy Work Mat, a set of counters, one dry-erase marker, and one dry-erase eraser to each student.Read the following scenario to the class: Tyra baked 48 delicious sugar cookies for the school bake sale. When she got ready to put an equal number of cookies in bags, she realized she only had 8 bags. How many cookies must she be able to put in each bag?.... The first strategy is the equal groups and shares strategy. Instruct students to count out 48 counters. Explain that in order to use the equal groups and shares strategy, we need to equally share the dividend amount of 48 counters among 8 groups.Under “Division Models” on the Strategy Work Mat, instruct students to draw 8 large circles to represent the groups. Place one counter in each of the 8 circles. Instruct students to repeat this process until all 48 counters have been placed in the circles.Discuss the following questions:(Point to a circle.) How many counters are in this circle? 6 (Point to another circle.) How many counters are in this circle? 6 Continue pointing to the remaining circles and asking students how many counters are in each circle.Point out that each circle contains an equal number of counters. Explain that when dividing an amount into groups, the amounts in each group have to be equal.Discuss the following question:How many cookies will Tyra need to put in each bag? 6”

• Scope 10: Multiplying Fractions by Whole Numbers, Intervention, Supplemental Aids, Procedure and Facilitation Points along with Print Files, Student Handout–1,2,3,4,6,8 provides for students’ active participation in content through the use of manipulatives. Procedure and Facilitation Points: “The Student Handouts contain a variety of fraction strips. These can be used to reinforce the following fraction concepts:Identifying fractionsExplaining fractional partsCounting fractional partsRepresenting fractionsComposing and decomposing fractionsFinding equivalent fractionsComparing fractionsAdding and subtracting fractionsMultiplying and dividing fractionsIf possible, provide a laminated copy of the fraction strips for each student. The students may then use dry-erase markers to shade and identify various fractions.Encourage students to draw fraction strips at the top of their paper or assessment as a reminder when working with fractions.” These instructions are used in conjunction with the Student Handout - 1,2,3,4,6,8 where fraction strips are provided for halves, thirds, fourths, sixths and eighths are provided but not labeled.”