4th Grade - Gateway 2

The instructional materials for Achievement First Mathematics Grade 4 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade level. For example: 

• In Unit 4, Lesson 8, students develop conceptual understanding of 4.NBT.5, as they use place value blocks to help them solve multi-digit multiplication problems. In Problem of the Day states, “Problem: A video store display shelf has DVDs stacked in 3 rows. There are 246 videos in each row. How many videos can the shelves hold? TT: How can we represent this problem with an equation? We can write 246 videos x 3 rows = K total videos. Add to VA. Why does that work? It works because this is a problem about equal groups. In this problem, we have 3 groups—the rows—with 246 DVDs in each. We need to figure out the total number of DVDs. We can do this with multiplication. Today we’re going to solve 2, 3, and 4-digit multiplication with place value blocks. Work with your partner to solve this equation with place value blocks.”
• In Unit 6, Lesson 6, students develop conceptual understanding of 4.NF.1, as they use tape diagrams and number lines to find equivalent fractions. In Workshop, Problem 2 states, “Markette is using a number line to figure out how many sixths are equal to $$\frac{2}{3}$$. She tells her partner, “We should partition each interval on our number line into 3 new parts, because $$3+3$$ is $$6$$. Is Markette’s strategy reasonable? Explain on the lines below. You may use pictures, number sentences, or number lines to help you.”
• In Unit 7, Lesson 6, students develop conceptual understanding of 4.NF.7, compare two decimals to the hundredths by reasoning about their size. In the Workshop, Problem 1, students use visual models and number lines to support their reasoning about comparisons. The materials state, “For each problem, shade each decimal amount on the given grids and plot them on the number line. Then use those models to compare the decimals using <, > or $$=0.2$$__$$0.19$$.” 

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. For example: 

• In Unit 2, Lesson 4, students demonstrate conceptual understanding of 4.NBT.2, as they use a provided place value chart and their knowledge of place value to determine the reasonableness of a provided answer. In Independent Practice, Problem 2 states, “Kate used the place value chart to write the number below in standard form. Is Kate’s work correct? Explain why or why not on the lines below.”
• In Unit 6, Lesson 1, students demonstrate conceptual understanding of 4.NF.3, as they draw a visual model and write an equation to solve a problem. In Independent Practice, Problem 1 states, “Terrell is keeping track of his running for the week. Draw a visual model and write an addition equation to model Terrell’s running plan. How far will he have to run at the end of the week?” 
• In Unit 10, Lesson 3, students demonstrate conceptual understanding of 4.MD.5, as they use manipulatives to find the measure of a given angle. In the Exit Ticket, Problem 3 states, “Using pattern blocks, how can you find the measure of the angle below? Use pictures, words and numbers to show how you found your answer.”

The instructional materials for Achievement First Mathematics Grade 4 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. The instructional materials include opportunities for students to build procedural skill and fluency in both Math Practice and Cumulative Review worksheets. The materials do not include collaborative or independent games, math center activities, or non-paper/pencil activities to develop procedural skill and fluency.

Math Practice is intended to “build procedural skill and fluency” and occurs four days a week for 10 minutes. There are eight Practice Workbooks in Achievement First Mathematics, Grade 3. One workbook, C, contains resources to support the procedural skill and fluency standard 4:NBT.4: Fluently add and subtract multi-digit whole numbers using the standard algorithm. In the Guide To Implementing Achievement First Mathematics Grade 4, teachers are provided with guidance for which workbook to use based on the unit of instruction. For example:

• In Practice Workbook C, Problem 1, students solve subtraction problems. It states, “Find the difference. 51,348 and 22,122. Use the standard algorithm to solve.” (4.NBT.4)
• In Practice Workbook C, Problem 6, students solve subtraction problems. It states, “Use a strategy that makes sense to you to solve. 59,637 – 34,721 = .” (4.NBT.4)
• In Practice Workbook C, Problem 9, students practice subtraction. The problem states, “$$56432 - 33224 =$$ _____.“ (4.NBT.4)

Cumulative Reviews are intended to “facilitate the making of connections and build fluency or solidify understandings of the skills and concepts students have acquired throughout the week to strategically revisit concepts, mostly focused on major work of the grade.” Cumulative Review occurs every Friday for 20 minutes. For example:

• In Unit 4, Cumulative Review 4.5, Problem 4, students solve subtraction problems. It states, “Find the difference. Show your work. $$6,241 - 1,366 =$$ _____.” (4.NBT.4) 
• In Unit 4, Cumulative Review 4.5, Problem 2, students solve addition and subtraction problems using the standard algorithm. It states, “Camden Yard sold 5,864 tickets and 2,549 students tickets to last Friday's Baltimore Orioles baseball game. How many total tickets were sold for last Friday’s game?”
• In Unit 6, Cumulative Review 6.4, Problem 5, students solve a subtraction problem. It states, “Find the difference. $$2,301-1,976 =$$ ___” (4.NBT.4)

The instructional materials for Achievement First Mathematics Grade 4 meet expectations for balancing the three aspects of rigor. Overall, within the instructional materials the three aspects of rigor are not always treated together and are not always treated separately.

The instructional materials include opportunities for students to independently demonstrate the three aspects of rigor. For example:

• In Unit 2, Cumulative Review 2.2, students demonstrate conceptual understanding of factors by determining whether there are additional factors for a number within 100. Problem 7 states, “Marco and Desiree made 56 cookies for a bake sale. They will put an equal amount of cookies into bags. Marco and Desiree want to put more than 2 cookies but fewer than 10 cookies into each bag. Desiree says that they can only put 7 cookies into 8 bags or 8 cookies into 7 bags. Marco thinks there are more ways to put an equal number of cookies into bags. Who is right? Why are they right?” (4.OA.4)
• In Practice Workbook B, students develop procedural skill and fluency as they multiply whole numbers. Problem 14 states, “Calculate the product of $$64×35$$.”  (4.NBT.5) 
• In Unit 6, Lesson 23, Exit Ticket, students apply their understanding of fraction multiplication as they solve word problems. The materials state, “Edwin uses $$\frac{3}{4}$$ of a teaspoon of baking powder for each batch of muffins he makes. He needs to make 3 batches for his Cub Scout meeting and 4 batches for his study group. How many teaspoons of baking powder will Edwin need?” (4.NF.4) 

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

• In Unit 4, Lesson 11, Independent Practice, students apply their conceptual understanding of adding decimals to solve a real-world problem. The materials state, “Mrs. Evans, the physical education teacher, is forming relay teams to help raise money for cancer research. There must be two students on each relay team. To determine the teams, Ms. Evans uses the students’ practice times from the last physical education class. Ms. Evans wants the teams to be as evenly matched as possible so they have a fair chance to win the race. What would the best combination of students be for each of the relay teams? Show all your mathematical thinking.” (4.NF.5, 4.NF.7) 
• In Unit 5, Lesson 7, Exit Ticket, students apply their conceptual understanding of multiplication to solve a two-step word problem using tape diagrams and equations. Problem 1 states, “Draw a tape diagram to model the following equation. Create a word problem. Solve for the value of the variable. $$(A×2)+4,892=6,392$$.” (4.OA.3) 
• In Unit 8, Lesson 4, Independent Practices, students apply their conceptual understanding of place value to solve a problem involving the value of coins. Problem 6 states, “Which is more, 68 dimes or 679 pennies? Prove with a place value chart and then explain on the lines below.” (4.MD.2)

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

The instructional materials provide opportunities for students to engage in constructing viable arguments and analyzing the work and understanding of others. Examples of opportunities for students to construct viable arguments and/or critique the reasoning of others include:

• In Unit 2, Lesson 16, Workshop, students critique the reasoning of others and construct a viable argument as they evaluate the estimation strategies used by other students to determine who is correct. Problem 2 states, “Patricia said the best way to estimate the solution to $$8,421 - 462$$ is to round each number to the nearest hundred. Matthew said the best way to estimate is to round each number to the nearest thousand. Who is correct? Explain your answer.” 
• In Unit 7, Lesson 4, Exit Ticket, students use their knowledge of decimal comparisons to the hundredths place to critique the reasoning of others and construct a viable argument. Problem 3 states, “Patrice is measuring the rainfall for December. On Monday there was 0.09 of an inch of rainfall. On Tuesday there was 0.9 of an inch of rainfall. Patrice tells his sister that it rained the same amount on Monday and Tuesday. Tell whether or not Patrice is correct on the lines below.”
• In Unit 10, Lesson 4, Exit Ticket, students construct an argument and critique the reasoning of others based on their knowledge of shapes. Problem 3 states, “Carlos is helping his brother with homework. He tells his brother that if you want to draw an obtuse angle, you should always use the bottom set of degrees on the protractor arc, and if you want to draw an acute angle you should always use the top set. Is Carlos’s reasoning accurate? Explain why or why not on the lines below.”

The instructional materials reviewed for Achievement First Mathematics Grade 4 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. There are opportunities within the teacher materials that guide teachers in assisting students to construct viable arguments and analyze the arguments of others, through the use of questioning.

Examples of the instructional materials assisting teachers in engaging students to construct viable arguments and analyze the arguments of others include:

• In Unit 2, Lesson 12, Problem of the Day, teachers are provided with language to assist students in analyzing the work of others. Try One More states, “Tonya is building numbers in math class. She wants to use her place value blocks to make some numbers that are greater than her partner’s number, 150,600. However, she only has one hundred-thousand block and no hundreds blocks. Write some numbers that Tonya could build that would be greater than her partner’s. Work with your partner to solve. Circulate as students work, look for scholars who are using place value charts or other efficient strategies to brainstorm numbers that fit the criteria. What strategy did you use to create numbers that fit the problem? We started by drawing a place value chart and writing a 1 in the hundred-thousands place and a 0 in the hundreds place, then we chose digits that were bigger than the ones in the partner’s number for all the other places. I have an idea, help me see if it works. Jot down ’98,023’. No! Why not? That number doesn’t work because it doesn’t have any hundred-thousands, so it's smaller than Tonya’s partner’s number because that number has a hundred- thousand in it, and even only 1 hundred-thousand is bigger than 9 ten-thousands! But 9 is more than 5! Yes, but the 9 is in the ten-thousands place and ten-thousands are less than hundred-thousands! What mistake was I making? You didn’t think about the largest place value!” 
• In Unit 7, Lesson 8, Discussion, teachers are provided with guidance and questions to engage students in critiquing the reasoning of others. The materials state, “Last year I had a scholar who told me that when you compare decimals it's like the opposite of comparing whole numbers. TT: What do you think they meant by that? They probably meant that when you compare whole numbers, tens are smaller than hundreds, but with decimals tenths are bigger than hundredths. They probably meant that when you are comparing whole numbers, the more digits you have the larger the number is, but that isn’t always true with decimals, like with 0.8 and 0.09. Yes! How are comparing whole numbers and comparing decimals similar? For both we have to think about the place value that digits are in.”
• In Unit 7, Math Stories, students are provided with an opportunity to share their math thinking as they solve problems involving fractions. Problem 12 states, “Yanira ran $$\frac{3}{4}$$ miles each day for 6 days. How many miles did she run over the course of 6 days?” Teachers are provided with three specific protocols to assist them in helping students represent and/or solve the problem, including sentence stems, for example: “First I put ____ because the story ____. Then I put ____ because in the story ____. Finally, I put ____ because in the story/we need to figure out ____.”

The instructional materials reviewed for Achievement First Mathematics Grade 4 meet  expectations that materials explicitly attend to the specialized language of mathematics. 

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics, and support students in using them. 

Examples of explicit instruction on the use of mathematical language include:

• In Unit 4, Lesson 15, teachers are provided with instructions to explicitly teach the term remainder. Try One More states, “We do have 1 leftover in this problem. This is called our remainder. A remainder is the amount leftover after dividing a number when one number does not divide evenly into another number. Reveal KP on VA. What was our answer before the remainder and why? 114 because we have 1 hundred + 1 ten + 4 ones. Add to VA. Yes. Now our answer becomes 114 R 1 because our answer is 114 with a remainder of 1.”
• In Unit 6 Lesson 1, teachers are provided with guidance in reviewing vocabulary related to fractions. The introduction states, “Before you get started, let’s review some key fraction vocabulary. In a fraction, what does the denominator tell us? The denominator tells us the total number of parts in a whole. In a fraction, what does the numerator tell us? The numerator tells us the total number of parts being referred to.”
• In Unit 9, Lesson 4, provides teachers with guidance in introducing terminology related to triangles through a series of questions. The introduction states, “We have learned about different angle types, which will help us in our work today. What are the different types of angles? Today you will use the different types of angles to help you classify triangles!” Students work to observe and note information about triangles. The materials state, “What did you observe about triangles? Triangles are classified by two names, kind of like how you have a first name and a last name. One name tells us about their angles, and one name tells us about their angles, and one name tells us about their sides.” 

Examples of the materials using precise and accurate terminology and definitions in student materials:

• In Unit 2, Lesson 1, Exit Ticket, students solve problems requiring them to demonstrate an understanding of the term expanded form. Problem 3 states, “What two hundreds is seven hundred twelve between? Write seven hundred twelve in standard form. Write seven hundred twelve in expanded form.” 
• In Unit 6, Lesson 8, Exit Ticket, accurate terminology is used as students are expected to partition shapes and compare their size. Problem 1 states, “Partition and label the number line below to show the following fractions. Then write TWO inequalities to compare two of the fractions.  $$\frac{2}{3}$$, $$\frac{1}{6}$$, $$\frac{4}{6}$$, Inequality 1: Inequality 2:.”
• In Unit 8, Lesson 2, Independent Practice, accurate terminology is used as students solve problems related to volume and capacity. Problem 1 states, “The capacity of each pitcher in the teacher work room is 3 quarts. Right now, each pitcher contains 1 quart 3 cups of liquid. If there are 3 pitchers in the room, how much more total liquid can the pitchers hold?”