5th Grade - Gateway 2

The instructional materials for Achievement First Mathematics Grade 5 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 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. For example, Partner Practice, Problem 1 states, “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.”
• In 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 materials state, “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.”  
• In 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 materials state, “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. For example: 

• In 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 states, “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.”
• In 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. In the Independent Practice, Bachelor Level, Problem 2 states, “Use the decimal grid below to solve: $$0.81-0.16=$$ ?” 
• In Unit 8, Lesson 7, students develop conceptual understanding of 5.NF.4, as they create area models to multiply unit fractions. In the Independent Practice, Bachelor Level, Problem 3 states, “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 instructional materials for Achievement First Mathematics Grade 5 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 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 instructional materials develop procedural skill and fluency throughout the grade level. For example:

• In 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 states, “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) 
• In Unit 3, Lesson 8, students reflect upon and choose an appropriate strategy for multiplication. Think About It states, “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) 
• In Unit 5, Mixed Practice 5.1, students develop procedural skill and fluency related to multiplication as they solve a word problem. Problem 3 states, “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 instructional materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. For example: 

• In 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.  The materials state, “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)
• In Unit 4, Skill Fluency 4.2, Day 2, students demonstrate fluency in multiplying multi-digit whole numbers using the standard algorithm. Problem 1 states, “Find the product of 736 and 92.” (5.NBT.5)  
• In Unit 7, Skill Fluency 7.1, Day 3, students demonstrate procedural skill and fluency with multiplication. Problem 3 states, “$$62×?=5952$$. Find the value of ?.” (5.NBT.5)

The instructional materials for Achievement First Mathematics Grade 5 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 4, Mixed Practice 4.1, students develop procedural skill and fluency as they solve problems involving multi-digit multiplication. Problem 3 states, “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) 
• In Unit 5, Cumulative Review, Problem of the Day, Day 2, students apply skills related to measurement conversions as they solve a routine problem. The materials state, “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) 
• In 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. In Problem 3, students are shown a 100 grid with two rows of 10 filled in. The materials state, “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. For example:

• In 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 states, “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)
• In 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 states, “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) 
• In 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 states, “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 instructional materials reviewed for Achievement First Mathematics Grade 5 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

The materials provide students with the opportunity to critique the work of others and engage them in extending their thinking to justify their responses. Examples of opportunities for students to construct viable arguments and/or critique the reasoning of others include:

• In 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 states, “Paul divided 8,280 by 36 and got 23. Do you agree or disagree? Prove your thinking and explain in the space below.”
• In Unit 5, Lesson 3, Independent Practice, Bachelor Level, students construct a viable argument and critique the reasoning of others based on their knowledge of multiplication. Problem 3 states, “The Town of Andover has an annual road running race that is 5,000 meters long. Alexis and Keith did mental math to determine the length of the race in kilometers. Which runner’s thinking is correct? Explain. Alexis’s Thinking $$5000 × 1000 = 5,000,000$$ km, Keith’s Thinking $$5000÷1000 = 5$$ km.”
• In 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 states, “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?” 
• In Unit 7, Lesson 11, Day 2, Partner Practice, Bachelor Level, students construct an argument based on their knowledge of fractions. Problem 1 states, “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.” 
• In 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 states, “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.” 
• In Unit 10, Lesson 9, Interaction With New Material, students critique the reasoning of others as they classify triangles. The materials state, “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 instructional materials reviewed for Achievement First Mathematics Grade 5 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. The materials provide teachers guidance to engage students in both constructing viable arguments and analyzing the arguments of others across the mathematics of the grade-level. For example: 

• In Unit 3, Lesson 3, guidance is provided to teachers in the form of a think-aloud exemplar to help students compare fictional students’ strategies. Think About It, Debrief states, “Which of these strategies was better in this problem? Using compatible numbers was better because they were closer to the actual numbers, so the estimate will be much closer to the real cost.” 
• In Unit 4, Lesson 2, Day 2, Debrief, provides teachers with guidance to support engaging students in analyzing the work of others in an error analysis problem.
  The materials state, “See the error: Which student’s work did you DISAGREE with most? Vote. Turn and tell your partner who you chose and why. Revote. CC. I see you disagree with student A. Why do you disagree with student A? SMS: I disagree with student A’s work because 500 and 70 are not compatible numbers. You can’t divide 500 by 70 mentally, it doesn’t include a basic fact. BPQ: What is missing from Student A’s number sentence that would make it possible to mentally divide? SMS: A basic fact they are not compatible numbers (NOTE: Give students the vocab compatible numbers as values that include a basic fact making them easy to compute mentally, if they do not remember from yesterday.) BPQ: Why is it important to round to compatible numbers when estimating quotients? SMS: Estimation is supposed to be done mentally and if you don’t round to compatible numbers that are a basic fact, it is hard to compute mentally. What error did this student make? CC. SMS: S/he rounded to the highest place value instead of to compatible numbers that included a basic fact.Let’s cross this one off.” 
• In Unit 8, Lesson 5, guidance is provided to teachers in the form of language to support them in helping students compare fictional student’s work and critique. Talk About It states, “Which do you agree with ...Vote...You agree with scholar B, which means you disagree with scholar A. Why do we disagree with Scholar A?” 
• In Unit 11, Lesson 5, Day 2, the materials guide the teacher to support engaging students in analyzing the reasoning of others in a real world word problem. The Debrief states, “See the Error: Which student’s explanation did you agree with? Vote. Turn and tell your partner who you chose and why. (Listening for exemplar understandings during TT). Revote. CC. SMS: Student B is correct. Both students chose the correct ordered pair, but Student B read the label for the x-axis carefully and saw that it did not show hours but instead showed actual times of day, so you had to calculate how many hours there are between 6 and 12.  What error did this student make? CC. SMS: Student A assumed that the coordinate was the number of hours that passed not the time that it was.”

The instructional materials reviewed for Achievement First Mathematics Grade 5 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 2, guidance is available for teachers to provide explicit instruction on the term compatible numbers. The debrief states, “We call numbers like 240 and 20 ‘compatible numbers’ in a division expression because together they contain basic facts that are easy to compute mentally, which is ONE requirement of estimation. But is this a valid method? How do you know? TT. CC. SMS: It is a valid method because it leads to an estimate that is close to the actual answer. The actual quotient is 7.219 when I use a calculator, and our estimate is 8 so that’s definitely in the same ballpark/reasonable.”
• In Unit 8, Lesson 12, provides teachers with guidance to introduce the new vocabulary term scaling. Think About It states, “When we want to increase or decrease an amount by a certain factor, it is called ‘scaling.’ You have probably solved problems before that asked you to find $$1\frac{1}{2}$$ times as much, or $$\frac{3}{4}$$ of an amount. The number we multiply by when doing this is called a ‘scale factor.’”
• In Unit 10, Lesson 8, provides strategies for teachers to use in guiding students to use precise vocabulary when classifying triangles. The debrief states, “Using the precise words for angles less than, equal to, or greater than 90, what name could we give each group, and why? TT. CC. SMS: Acute, Right, and Obtuse, because group 1 has only acute angles, group 2 has a right angle, and group 3 has an obtuse angle.What attribute are we using to sort these? CC. SMS: The measure of the angles.”

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

• In Unit 3, Lesson 2, Independent Practice, Bachelors Level, accurate terminology is used as students identify expressions. Problem 2 states, “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.”
• In Unit 5, Lesson 4, Independent Practice, Bachelors Level, students are expected to understand and use accurate terminology as they solve a division problem and explain their answer. Problem 5 states, “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.”
• In Unit 8, Lesson 7, Exit Ticket, accurate terminology is used as students create an area model to solve a problem. Problem 1 states, “What is $$\frac{1}{2}$$ of $$\frac{2}{5}$$? Part A. Use an area model to solve. Part B. Why does it make sense that your product is less than $$\frac{2}{5}$$?”