5th Grade - Gateway 2

The instructional materials for Achievement First Mathematics Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade level. Examples include: 

• Unit 3, Lesson 4, students develop conceptual understanding of 5.NBT.5, as they calculate products of two- and three-digit numbers by one-digit factors using area models. Partner Practice, Problem 1, “Taliyah’s brother sells 654 gallons of cookie dough for $7 each. How much money does her brother raise? a) Find the product using the distributive property and an area model.” (a partially filled out area model is provided) “b) Use the standard algorithm to solve the multiplication problem. c) Describe each of the partial products you calculated, in order, when using the standard algorithm.”

• Unit 6, Lesson 4, students develop conceptual understanding of 5.MD.5, as they use visual models of shapes to write expressions related to volume. In the Independent Practice, Bachelor Level, Problem 1, provides students with a 4×4×5 rectangular prism. “The same prism is shown below three times. Each cube represents one cubic meter. On each prism, use the lines to show you how you can deconstruct it into layers in a different way. Then, below each prism, write an expression to find the volume of each prism and solve.” 

• Unit 8, Lesson 2, students develop conceptual understanding of 5.NF.3, as they use tape diagrams to solve division problems. In Think About It, students are introduced to tape models to solve, “8 ÷ 4 = and 3 ÷ 4 = . The models below are called tape diagrams. Part A. Use the models provided to determine each quotient. Circle the quotation in your model. Part B. In the space below each model, show a check step to prove that each quotient is correct.”

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

• Unit 2, Mixed Practice 2.1, students demonstrate conceptual understanding of 5.NBT.A, as they explain patterns in products when multiplying by powers of ten. Problem 2, “Matthew multiplied 1.5×10^3 and said that the answer was 1.5000. Which statement, if any, explains Matthew’s error? a. Matthew multiplied 10 by the exponent 3 b. Matthew multiplied 1.5 by the exponent 3 c. Matthew added 3 zeroes to the end of 1.5 d. Matthew’s statement is correct and contains no errors.”

• Unit 7, Lesson 1, students demonstrate conceptual understanding of 5.NBT.7, as they use a decimal grid to solve a subtraction problem involving decimals. Independent Practice, Bachelor Level, Problem 2, “Use the decimal grid below to solve: 0.81-0.16=?” 

• Unit 8, Lesson 7, students develop conceptual understanding of 5.NF.4, as they create area models to multiply unit fractions. Independent Practice, Bachelor Level, Problem 3, “What is the area of a rectangle that is \frac{1}{2} yard long and \frac{3}{8} yard wide? A 1 by 1 yard rectangle has been started for you below.”

The materials for Achievement First Mathematics Grade 5 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The materials include opportunities for students to build procedural skill and fluency in both Skill Fluency and Cumulative Review (Mixed Practice) components. 

The publisher states that the Skill Fluency component of the curriculum “addresses the skill, procedures and concepts that students must perform quickly and accurately in order to master a standard or a skill imbedded within a standard. Skill Fluency is delivered during a 10-minutes segment of a 90-minute period.” The Skill Fluency and Cumulative Review (Mixed Practice) components contain resources to support the procedural skill and fluency standard 5.NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm. 

The materials develop procedural skill and fluency throughout the grade level. Examples include:

• Unit 3, Lesson 4, Independent Practice, Bachelor Level, students estimate and connect partial products to the standard algorithm as they multiply a one-digit number by a three-digit number. Problem 3, “For each problem, make an estimate first. Then calculate the product using the standard algorithm and show your work. For number 3, list each of the partial products being calculated in order as shown in number 1. Use estimation to check the reasonableness of your product: 464 × 5 = ____.” (5.NBT.5) 

• Unit 3, Lesson 8, students reflect upon and choose an appropriate strategy for multiplication. Think About It, “We’ve studied several methods for multiplying in this unit and in previous grades, including mental math, the distributive property (with an area model or expression) and the standard algorithm. Look at each problem below and decide which of these strategies makes the most sense to use.” Students solve, “$$7 × 8$$, 85 × 10, 5 × 17, and 422 × 329” (5.NBT.5) 

• Unit 5, Mixed Practice 5.1, students develop procedural skill and fluency related to multiplication as they solve a word problem. Problem 3, “Over the course of fifteen days, a museum counts the number of guests that enter. They count an average of 2,362 people on each of the days. How many guests visited the museum altogether. Show your work. Answer ________.” (5.NBT.5)

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

• Unit 3, Mixed Practice, 3.2, Day 2, students demonstrate procedural skill and fluency as they multiply multi-digit factors while solving a problem with a provided chart. “Rory, Elaina and Yashika are all on a reading marathon team. The time each girl reads each day is shown in the chart below. If each girl reads for 36 days, how many total minutes will they have read?” (5.NBT.5)

• Unit 4, Skill Fluency 4.2, Day 2, students demonstrate fluency in multiplying multi-digit whole numbers using the standard algorithm. Problem 1, “Find the product of 736 and 92.” (5.NBT.5) 

• Unit 7, Skill Fluency 7.1, Day 3, students demonstrate procedural skill and fluency with multiplication. Problem 3, “$$62 × ? = 5952$$. Find the value of ?.” (5.NBT.5)

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Students are given multiple opportunities to engage in real-world applications especially within exercise based lessons as well as the problem of the day in each cumulative review. 

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 2, Lesson 1, Mixed Practice Day 2, Problem 5, students solve a real-world, non-routine problem by comparing two decimals to thousandths based on meanings of the digits in each place. (5.NBT.3) "Noah threw a Frisbee 4.89 yards. a) Noah threw the Frisbee farther than Lin. How far could Lin have thrown the Frisbee? b) Andre threw the Frisbee farther than Noah but less than 4.9 yards. How far could Andre have thrown the Frisbee? Explain your reasoning.”

• Unit 3, Lesson 3, Problem of the Day, Day 1, students write and interpret an expression then solve a routine real-world problem involving multiplying multi-digit whole numbers using the standard algorithm (5.NBT.5). "Use the chart to solve. (note: the chart shows minimum and maximum length and width of High School and FIFA Regulation Soccer Field Dimensions) a. Write an expression to find the difference in the maximum area and minimum area of NYS high school soccer fields. Then, evaluate your expression. b. Would a field with a width of 75 yards and an area of 7,500 square yards be within FIFA regulation? Explain why or why not.”

• Unit 7, Lesson 11, Interaction with New Material, students engage in a routine problem with 5.NF.1 as they add and subtract fractions with unlike denominators. "Victor is making a special enchilada dish for the Latin Heritage festival at his school. To make the dish, he needs a lot of fresh tomatillos. To make enough for 60 servings he needs 12\frac{1}{2} pounds of tomatillos. He finds 5\frac{1}{4} pounds at King’s Grocery and 3\frac{3}{5} pounds at Metropolitan Grocers. He decides to call a third store to see if they’ll have enough in stock. How much should he ask for?”

• Unit 9, Lesson 3, Think About It, students engage with 5.NF.7 as they apply and extend previous understanding of division to divide unit fractions by whole numbers and whole numbers by unit fractions in a non-routine problem. Think About It, “Carmine and Miguel are working together on the following problem: Mrs. Silverstein is having a college graduation party for her son. She buys enough cake so that each guest at the party can have up to \frac{1}{6} of a cake. She buys 3 cakes. How many guests is she expecting? Carmine writes the equation \frac{1}{6} ÷ 3 = \frac{1}{18}. Miguel writes the equation 3 ÷ \frac{1}{6} = 18. Is either student correct? Create a model to prove your thinking. Then explain your reasoning on the lines below.”

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

• Unit 6, Lesson 10, Independent Practice Question 1 (Bachelor Level), students apply the volume formula (5.MD.5) and convert among different-sized standard measurement units within a given measurement system (5.MD.1) in the context of solving a non-routine real-world problem. "At the flea market, a shopper asks Geoffrey if it is possible to use his 3 foot by 2 foot by 2-foot large planter as a bookcase or storage instead. Geoffrey considers this and estimates that a typical book has a volume of about 40 cubic inches. How many books would a large planter hold if filled with as many books as possible?”

• Unit 9, Lesson 3, Independent Practice Question 2 (Bachelor Level), students engage with 5.NF.7 as they solve a routine real-world problem involving division of unit fractions. "Virgil has \frac{1}{6} of a birthday cake left over. He wants to share the leftover cake with 3 friends. What fraction of the original cake will each of the 3 people receive? Draw a picture to support your response.”

• Unit 11, Lesson 5, Real World Problems, students represent routine real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation (5.G.2). "There is a $25 annual fee for membership at the gym. It also costs $5 per visit to use the gym. Fill in the table to show the total cost of x visits to the gym. a) Write the ordered pairs, and graph the data on the coordinate graph. b) Write the ordered pair that represents 6 visits to the gym. Explain what the ordered pair means. c) If Amaya can only spend up to $50 in one month, how many times can she visit the gym? Explain.”

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

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

• Unit 4, Mixed Practice 4.1, students develop procedural skill and fluency as they solve problems involving multi-digit multiplication. Problem 3, “Find a 3-digit number and a 1-digit number that when multiplied together will result in a product between 3,000 and 4,000. Show your work.” (5.NBT.5) 

• Unit 5, Cumulative Review, Problem of the Day, Day 2, students apply skills related to measurement conversions as they solve a routine problem. “A city wants to install fencing around two new playgrounds. Playground A is 5 yards long and 25 feet wide. Playground B is 3 yards long and 27 feet wide. A) Which playground will require more fencing, and by how much? B) Fencing costs $15 per two feet. How much will it cost to put up fencing around both playgrounds?” (5.MD.1) 

• Unit 7, Lesson 2, Independent Practice, Bachelor Level students develop conceptual understanding of adding and subtracting decimals to the hundredths as they use a hundreds grid to solve a problem. Problem 3, students are shown a 100 grid with two rows of 10 filled in. “Jonah added 0.36 to the value below and got 2.36. Is his answer reasonable? Why or why not? (Use the space to the right to explain.)” (5.NBT.7)

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

• Unit 3, Lesson 4, Partner Practice, students develop conceptual understanding of place value and procedural skills and fluency as they solve a problem involving the standard algorithm, to find a product in a real world context. Problem 1, “Taliyah’s brother sells 654 gallons of cookie dough for $7 each. How much money does her brother raise? a) Find the product using the distributive property and an area model. b) Use the standard algorithm to solve the multiplication problem.” (5.NBT.5)

• Unit 7, Lesson 12. Independent Practice, Master Level, students develop conceptual understanding of fractions and apply skills related to addition and subtraction of fractions as they solve a problem and develop a model. Problem 1, “Directions: Create a model of both scenarios. Write an equation that could be used to find a solution in each scenario. Explain how the scenarios are similar and how they are different. Problem A: Jennah has one piece of string that is 3\frac{1}{8} meters long, and another that is 3\frac{5}{10} meter. How much longer is the longer string? Model: ____, Equation: ____ . Problem B: Jennah had a piece of string that was 3\frac{5}{10} meters long. She used 3\frac{1}{8} meters. How much string was left? Model: ____ Equation: ___. How are the problem scenarios mathematically similar? What is one important difference in the problem scenarios?” (5.NF.1, 5.NF.2) 

• Unit 8, Lesson 18, Independent Practice, Masters Level, students apply their understanding of fractions as they solve problems involving multiplication of fractions and mixed numbers, and demonstrate procedural skill to add and subtract decimals to hundredths. Problem 1, “Oliver came home from the store with .250 L of heavy cream only to find that he needed 1\frac{1}{3} times that much for his recipe. How much more heavy cream does he need when he goes back to the store? Represent the problem with a model and an expression or equation. Then solve.” (5.NF.6, 5.NBT.7)

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson. There are instances where the Unit Overview gives a detailed explanation of the MPs being addressed within the unit, but the lessons do not cite the same MPs.

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Examples include:

• The Unit 6 Overview outlines the intentional development of MP1. “In lesson 6, students work to make sense of problems by identifying unknowns in various volume-related contexts. In lessons 9 and 10, students make sense of complex volume problems in various contexts, persevering to properly formulate solution pathways and solutions.”

• Unit 7, Lesson 12, Partner Practice Question 1 (Bachelor Level), students make sense of how equations connect to verbal descriptions. “Which equation or equations can be used to represent the following: Team A built a tower that was 1\frac{1}{2} feet taller than Team B’s. Team B’s tower was 3\frac{1}{5} tall. How tall is Team A’s tower? Create a model, and then circle all equations that apply. a) 1\frac{1}{2} - \frac{1}{5} =?; b) 1\frac{1}{2} + ? = 3\frac{1}{5}; c) 3\frac{1}{5} - 1\frac{1}{2} =?; d) 3\frac{1}{5} + 1\frac{1}{2} = ?”

• The Unit 9 Overview describes development of MP1. “In both lessons 5 and 12, students extend their understandings of division by making sense of and persevering in solving multi-step problems in real world contexts. In lesson 5 students do this with the division of fractions, in lesson 12 with all operations of decimals.”

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

• Unit 4, Lesson 8, Interaction With New Material, students reason abstractly and quantitatively when working with dimensions of a room. “The owner of the art gallery knows that his rectangular space is 1500 square feet. The width of the space is 60 feet. In order to plan for a new exhibit, he needs to know the full perimeter of the gallery. Help him find it in the space below.”

• Unit 7, Lesson 13 Check for Understanding, students engage with MP2 as they solve a real world problem involving addition and subtraction of fractions with different denominators. “John got directions to his new high school for orientation day. He knows the school is 9 miles away. When he pulls the directions from his pocket, some of the last step was rubbed off. ~Take Atlantic Ave. 3\frac{1}{4} miles. Turn right on 17th. ~Go 2\frac{1}{8} miles. The road becomes Jefferson Ave. ~Take Jefferson Ave .... If the school is on Jefferson Ave., how many miles should John be on Jefferson Ave.? Draw a model and write an equation to represent the scenario.”

• The Unit 8 Overview outlines the intentional development of MP2. “In lesson 1-3, students relate fractions to division in different contexts. By de- and re-contextualizing fractions in these contexts students reason abstractly and quantitatively about the situations. In lessons 16 and 17, students reason quantitatively about the placement of their decimal in a product based on their estimates.”

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Examples include:

• The Guide to Implementing AF Math describes Error Analysis lessons as one way to address MP3. “Purpose: Through the use of error analysis, guided questioning and discussion students will identify and fix a common misconception related to a skill they learned the previous day. These are sequenced so that after a particularly complex conceptual lesson or a lesson involving a skill that surfaces a common misconception, students get another focused at bat to either fix their misunderstanding or deepen their reasoning around key mathematical concepts and viable strategies to guide them away from making the same error. These lessons start with analyzing fictional student work and are structurally based off of the Standards for Mathematical Practice 3.”

• Unit 4, Lesson 7, Partner Practice, Masters Level, students critique the reasoning of others and construct an argument based on their knowledge of division. Problem 1, “Paul divided 8,280 by 36 and got 23. Do you agree or disagree? Prove your thinking and explain in the space below.”

• Unit 6, Lesson 2, Independent Practice, Bachelor Level, students critique the reasoning of others and construct an argument based on their knowledge of shapes. Problem 7, “Tyler builds the shape below and then turns it on its side. He says that the figure takes up less space now because it is shorter. Do you agree or disagree with his claim and why?” 

• Unit 7, Lesson 11, Day 2, Partner Practice, Bachelor Level, students construct an argument based on their knowledge of fractions. Problem 1, “Which of the following differences will require regrouping to solve? 1\frac{1}{3} -\frac{1}{2} OR 1\frac{1}{2} -\frac{1}{3} Explain how you know without doing any calculations.” 

• Unit 8, Lesson 17, Day 2, Exit Ticket, students critique the reasoning of others as they use estimation to assess the reasonableness of an answer. Problem 2, “Tyler multiplies 3.1 and 4.2. He gets a product of 130.2. Using estimation as your evidence explain if his product is reasonable or unreasonable and what his mistake might have been.” 

• Unit 10, Lesson 9, Interaction With New Material, students critique the reasoning of others as they classify triangles. “Ms. Cox’s class is analyzing the two figures below. Mya says that they can be given the same name. Justin says the shapes have different names. Ms. Cox says that both students are correct. Part A. How is it possible that both students are correct? Explain your reasoning. Part B. What is the most specific name that can be given to each triangle? Justify your response.”

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

There is intentional development of MP6 to meet its full intent in connection to grade-level content. Many problems present students with the opportunity to attend to precision within the mathematics and the reasoning of the answer. Examples include: 

• The Unit 4 Overview, "In lessons 3-7 students attend to precision when they are transitioning their preferred division strategy to the written methods. This requires students to make connections to more concrete representations and to keep place values of quotients in order. MP 6 is a major focus of unit 4 as students must precisely identify place values when dividing throughout the unit."

• The Unit 6 Overview, “In lesson 1, students attend to precision by intentionally learning the proper expression of cubic units in volume and the why behind these three dimensions. This understanding is expanded upon in lesson 4 when students learn a different way of breaking down a 3D structure into layers. Finally in lesson 7, students learn of the additive nature of volume and how to define their units when adding multiple structures together. Students often attend to precision in lessons, however, in Unit 6, MP6 is specifically emphasized with a focus on the identification and tracking of proper units in volume contexts.”

• Unit 10, Lesson 8 provides strategies for teachers to use in guiding students to use precise vocabulary when classifying triangles. The debrief, “Using the precise words for angles less than, equal to, or greater than 90, what name could we give each group, and why?” Students might say, “Acute, Right, and Obtuse, because group 1 has only acute angles, group 2 has a right angle, and group 3 has an obtuse angle.”

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

• At the beginning of each lesson plan, there is a section labeled “Key Vocabulary” for the teacher. Unit 2, Lesson 2, Key Vocabulary,

• “Denominator – The bottom number in a fraction; shows the number of parts in the whole

• Numerator – The top number in a fraction; shows how many parts of the whole are being described

• Equivalent fraction – A fraction with the same value as another fraction but with different numerators and denominators.”

• Unit 3, Lesson 2, Independent Practice, Question 2 (Bachelors Level), accurate terminology is used as students identify expressions. “Which expression represents twice the product of 15 and 4? Circle all that apply. a. 2 + (15 × 4) b. 2 × (15 × 4) c. 2 × (15 + 4) d. 62 e. 120.”

Unit 5, Lesson 4, Independent Practice, Question 5 (Bachelors Level), students are expected to understand and use accurate terminology as they solve a division problem and explain their answer. “Myra converted 5,300 feet into miles using the correct expression 5,300 ÷ 5,280. She got a correct answer of 1 R20. What does the 1 in her quotient represent? What does the 20 represent? Explain.”

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 2, Lesson 4, Independent Practice, Question 1 (Bachelor Level), students engage with MP7 as they compare and order fractions. “Review: Compare the pairs of fractions by reasoning about the size of the units. Use >, <, or =. a) 1 fourth _____ 1 fifth; b) 3 fourths _____ 3 fifths; c) 1 tenth _____ 1 twelfth; d) 7 tenths _____ 7 twelfths.”

• Unit 10, Lesson 6, Think About It, students look for structure as they classify quadrilaterals. “Below each shape, list as many names as you can for the shape. Then, circle every name that they have in common.”

• Unit 11, Lesson 1, Exit Ticket, students interpret the structure of the coordinate plane as they construct a coordinate plane and use it to name the location of points. “Use a ruler on the grid below to construct the axes for a coordinate plane. The x-axis should intersect points L and M. Construct the y-axis so that it contains points K and L. Label each axis. a) Place a hash mark on each grid line on the x- and y-axis. b) Label each hash mark so that A is located at (1, 1). c) What are the coordinates of point M? d) What is point L called?”

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

• Unit 2, Lesson 1, Independent Practice, Question 2 (Bachelor Level), students repeatedly create equivalent fractions and make connections to multiplying and dividing by fractions equal to one. “Generate four fractions that have the same value as the fraction \frac{2}{5} . Show your work and record your answers on the line below.”

• Unit 7, Lesson 9, Day 2, Independent Practice, Question 3 (Bachelor Level), students find the least common denominator as an efficient shortcut or additional subtraction strategy with fractions. “Madame Curie made some radium in her lab. She used \frac{15}{36} kg of the radium in an experiment and had 1\frac{1}{18} kg left. Part A. How much radium did she have at first?”

Unit 10, Lesson 3, Test the Conjecture, Question 2, students use repeated reasoning to make sense of polygons by classifying quadrilaterals based on the presence of parallel sides. “True or false, a quadrilateral is always a trapezoid.”