2nd Grade - Gateway 2

The instructional materials reviewed for JUMP Math Grade 2 partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied; however, there are missed opportunities concerning the variety of problem types called for by the Standards.

The instructional materials provide students opportunities to engage in routine application of grade-level mathematics. The 2.OA.A cluster heading calls for students to “Represent and solve problems involving addition and subtraction.” Grade 2 standard 2.OA.1 calls for students to, “Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem” (see Table 1, CCSSM page 88). All problem types are not represented equally, as there is a missed opportunity for students to work with take apart addend unknown and two-step word problems in application situations.

Students are given multiple opportunities to practice representing addition and subtraction problems with drawings and equations with a symbol for the unknown number to represent the problem in routine applications. Take apart addend unknown and two-step word problems are underrepresented in the materials in application situations. Examples of word problems include:

• Teacher Resource, Part 2, Unit 1, Lesson OA2-46, Exercises, “a) Paul picks all the ripe tomatoes on a vine every day. On Monday there were 9 ripe tomatoes. Each day after, there are 2 fewer ripe tomatoes to pick. On which day will there be exactly 1 ripe tomato? b) How many ripe tomatoes did Paul pick in total?” (2.OA.1)
• Teacher Resource, Part 2, Unit 3, Lesson OA2-51, Exercises, “a) Ken put 7 paper clips into a bowl. He took out 3 paper clips. Then he put 5 more paper clips into the bowl. How many paper clips are in the bowl now? b) Randi bought 5 raffle tickets. She gave 4 tickets to her family. Then she bought 11 more tickets. How many tickets does she have now?” (2.OA.1)
• Teacher Resource, Part 2, Unit 3, Lesson OA2-59, “Pail has 3 more apples than Vera. Vera has 12 apples.” (2.OA.1)

The instructional materials have some opportunities for students to engage in non-routine application throughout the grade level. Examples of non-routine applications include:

• Teacher Resource, Part 1, Unit 2, Lesson OA2-17, Extension 2, “Jin writes an addition with three numbers: 5 + 9 + 3. a) Create a story problem about blocks of different colors that matches the addition. b) Use objects or pictures to find the answer to your story problem. c) Explain to a partner how your story problem matches Jin’s addition.” (2.OA.2)
• Teacher Resource, Part 1, Unit 8, Lesson MD2-13 Extension 3, “Clara has markers and erasers. The markers are each 9 cm long and the erasers are each 4 cm long. She puts the markers and erasers in a line end-to-end. The total length of the line is 30 cm. How many markers and erasers are in the line?” (2.MD.4,5)
• Teacher Resource, Part 2, Unit 1, Lesson OA2-47, Extension 5, “There are 13 red tulips and 9 roses in a garden. There are 4 more yellow tulips than red tulips. There are 6 fewer white tulips than yellow tulips. a) How many tulips of each color are there? Show your work and explain your solution. b) Which piece of information did you not need to use? Explain.” (2.OA.1)
• Teacher Resource, Part 2, Unit 3, Lesson OA2-51, Extensions 1-2, Students make a word problem for the number sentence in Extension 1, example, “16 - 7 + 2.” In Extension 2, students exchange the word problems they wrote in Extension 1 and answer each other’s problems. (2.OA.1)
• Teacher Resource, Part 2, Unit 5, Lesson MD2-5, Extension 1, students write a word problem for the subtraction sentence that is given and then give to a partner to answer using a part-whole picture. Example, “a) 20 total cm - 13 cm cut off = __ cm left.” (2.MD.5)

The instructional materials reviewed for JUMP Math Grade 2 partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The instructional materials address specific aspects of rigor, and the materials integrate aspects of rigor, however, not all aspects are addressed equally. Heavy emphasis is placed on conceptual understanding and procedural skill and fluency. While students are given opportunities to engage with application problems throughout the materials, these are often teacher directed.

All three aspects of rigor are present independently throughout the materials. Examples include:

• Conceptual Understanding: Teacher Resource, Part 2, Unit 2, Lesson NBT2-25, Activity 3, “Provide student pairs with hundreds, tens, and ones blocks. Students take turns writing or saying three-digit numbers for their partner to build with blocks. You may wish to have students draw their representations in their notebooks so that you can verify their work.”
• Procedural Skill and Fluency: Teacher Resource, Part 1, Unit 2, Lesson OA2-18, students add two one-digit numbers with sums greater than ten by first regrouping to make a 10. Students continue to practice this procedural skill and fluency with BLM Using 10 to Add (2) and in the accompanying Assessment and Practice Book pages.
• Application: Teacher Resource, Part 1, Unit 4, Lesson OA2-43, Extension, “Tony had 12 stickers. He gave some stickers to Kate. Then he gave some stickers to John. Now Tony has 6 stickers left. a) If Tony gave the same number of stickers to Kate and John, how many did he give to each person? b) If Tony gave Kate 2 more stickers than he gave John, how many stickers did he give to Kate?” 

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials; however, a heavy emphasis is placed on conceptual understanding and procedural skill and fluency. Examples of multiple aspects of rigor that are engaged simultaneously include:

• Teacher Resource, Part 1, Unit 6, Lesson NBT2-15, Exercises “Draw a number line to subtract. a) 47-44 b) 91-87.” Students demonstrate conceptual understanding and procedural skill and fluency as they draw and use a number line to subtract.
• Teacher Resource, Part 2, Unit 7, Lesson MD2-41, Extension 1, “Amir finds 27¢ in his backpack, 2 dimes and 4 pennies in his pocket, and 41¢ under his bed. He buys 2 limes that cost 33¢ each. How much money does he have now? Show your work and explain what each step means in the story problem.” Students demonstrate all aspects of rigor as they use strategies such as counting on and place value to count money and solve real world application problems within 100.
• Teacher Resource, Part 1, Unit 5, Lesson NBT2-11, Exercises: “Draw base ten diagrams for the addition. Group 10 ones to make a ten. Add the tens and then add the ones that are left. a) 26 + 9 b) 38 + 6 c) 54 + 7 Bonus: 145 + 8.” Students demonstrate conceptual understanding and procedural skill and fluency as they use base ten blocks and drawings to add a one-digit number to a two-digit number involving regrouping.

The instructional materials reviewed for JUMP Math Grade 2 partially meet expectations for carefully attending to the full meaning of each practice standard. The materials do not attend to the full meaning of MPs 5 and 7.

Examples of the materials carefully attending to the meaning of some MPs include: 

• MP1: Teacher Resource, Part 1, Unit 3, Lesson OA2-33, Extension 3,“List all the pairs of numbers that can fill in the blanks in the number sentence 3 + __ + __ = 11.” Students make sense of the problem and persevere in solving it when they must determine all of the possible combinations using three addends to reach a sum of 11. 
• MP2: Teacher Resource, Part 1, Unit 5, Lesson NBT2-8, Extension 2, “Tasha is a farmer. She has 27 cows and 43 pigs. She buys 30 animals from another farm. Now how many animals does she have? USe any tools you think will help and use number sentences to show your work and explain what each step means in the story.” Students reason abstractly and quantitatively when they use number sentences to represent real-world problems and explain each step.
• MP8: Teacher Resource, Part 2, Unit 9, Lesson MD2-47, Extension 3, “Look for a pattern in the numbers you make by starting at 2 and adding 5s. Use the pattern to write five 4-digit numbers that you get when you start at 2 and add 5s.” Students look for and express regularity in repeated reasoning when they find a pattern with numbers and use that pattern to make a four-digit number.

Examples of the materials not carefully attending to the meaning of MPs 5 and 7 include: 

• MP5: Teacher Resource, Part 2, Unit 4, Lesson NBT2-42, Extensions, Problem 5, “Use pencil and paper or mental math to answer the question. Explain your choice. a) 370+30.” Students are directed which tools to use, instead of selecting a tool strategically. 
• MP5: Teacher Resource, Part 2, Unit 5, Lesson MD2-18, Extensions, problem 1, “a) Estimate the length of your classroom. Use your fingers or meter sticks.” Students are directed which to use. 
• MP5: Teacher Resource, Part 2, Unit 6, Lesson MD2-34, “Tony started to practice piano at 4:15. He practiced for 20 minutes, took a break for 10 minutes, then practiced for 30 more minutes. What time did he finish? Use a clock or a timeline to solve the problem. Explain your solution.” Students are directed which tool to use, instead of selecting a tool strategically. 
• MP7: Teacher Resource, Part 2, Unit 2, Lesson NBT2-29, Extensions, Problem 5, “Find all the number words tha fit in the blanks and make the addition sentence true. Explain how you know you found all the answers. a) nine hundred + _ _ _ hundred = _ _ _ thousand.” Students do not use structure to solve problems, but instead are using spelling patterns to solve problems.
• MP7: Teacher Resource, Part 2, Unit 3, Lesson OA2-50, Extensions, Problem 1, “Skip count by two different numbers to find the total.” Students are given arrays of circles to skip count and do not use structure to solve problems.

The instructional materials reviewed for JUMP Math Grade 2 partially meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards. 

There are few opportunities in the Teacher Resource or the Assessment & Practice for students to construct viable arguments or analyze the arguments or the work of others. MP3 is identified in the margins of the lesson. Examples of where the materials prompt students to construct viable arguments or analyze the arguments of others include, but are not limited to:

• Teacher Resource, Part 1, Unit 5, Lesson NBT2-10, Extension 2, “a) Explain how you would find all the ways to add a two-digit number and a one-digit number to make 83. b) Write down all the ways to add a two-digit number and a one-digit number to make 83. c) Explain how you know you found all the ways. d) In pairs, share your explanations to part c). Do you agree with each other? Discuss why or why not.”
• Teacher Resource, Part 2, Unit 1, Lesson OA2-45, Extension 6, “A muffin can have up to 10 raisins and walnuts altogether. a) Will wants at least 2 more walnuts than raisins. How many of each can he have in his muffin? Explain how you know. b) Sam wants exactly 3 fewer walnuts than raisins. Can he get 10 walnuts and raisins in total? How do you know? c) In pairs, explain your answers to parts a) and b). Do you agree with each other? Discuss why or why not.”
• Teacher Resource, Part 1, Unit 2, Lesson OA2-25, Extension 5, “Sam has 15 balloons. 13 of them pop, so he throws them out. Then he buys 14 more balloons. a) Does Sam have more balloons at the first or at the end? Find a fast way to know. b) In pairs, explain your answer to part a). Do you agree with each other? Discuss why or why not. c) How many balloons does Sam have at the end? Explain how you know.” Students critique the reasoning of others when they agree or disagree with the method and answer of their partner for part a. Students construct an argument when they explain how they know the number of Sam’s balloons at the end.
• Teacher Resource, Part 1, Unit 7, Lesson MD2-3, Extension 2, “Amy measures the length of a pencil with small paper clips. Zara measures the same pencil using large paper clips. a) Rani says that Amy will get a smaller number of units than Zara because Amy uses a smaller unit of measurement. Do you agree with Rani? Explain. b) In pairs, explain your answers to part a). Do you agree with each other? Discuss why or why not.” Students analyze Rani’s reasoning about the number of units needed to measure a pencil and construct an argument to defend their thinking.
• Teacher Resource, Part 2, Unit 6, Lesson MD2-31, Extension 4, “Jo started at 0 and skip counted by 5s. All her numbers were even. a) Did Jo skip count correctly? How do you know? b) In pairs, explain your answers to part a). Do you agree with each other? Discuss why or why not.” Students analyze Jo’s results of her skip counting and construct an argument to defend their thinking of whether she skip counted correctly.

Examples where the materials miss opportunities to prompt students to construct viable arguments or analyze the arguments of others include, but are not limited to:

• Teacher Resource, Part 1, Unit 2, Lesson OA2-16, Extension 2, “a) A group of 50 people reserve dinner at a restaurant, so the restaurant needs to set the tables so that everyone gets a fork, a knife, and a plate. The restaurant has 48 plates, 53 knives, and 60 forks. Does the restaurant have enough plates, knives, and forks? b) In pairs, explain your answers to part a). Do you agree with each other? Discuss why or why not.” Explaining their answers does not mean students are creating mathematical arguments. 
• Teacher Resource, Part 1, Unit 2, Lesson OA2-19, Extension 3, “e) Guess how many numbers of numbers add to 18. f) In pairs, explain your answers to part e). Do you agree with each other? Discuss why or why not.” Before this problem, students find how many pairs of numbers add to 4, 5, 6, and 7. Explaining their answers does not mean students are creating mathematical arguments.

The instructional materials reviewed for JUMP Math Grade 2 partially meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. While students are given opportunities to construct viable arguments and analyze the reasoning of others, the materials provide limited assistance to teachers in engaging students in both constructing viable arguments and analyzing the arguments of others.

Teacher Resource, Part 1 states, “When students work with a partner, many of them will benefit from some guidance, such as displaying question or sentence stems on the board to encourage partners to understand and challenge each other’s thinking, use of vocabulary, or choice of tools or strategies. For example:

• I did ___ the same way but got a different answer. Let’s compare our work.
• What does __ mean?
• Why is __ true?
• Why do you think that__?
• I don’t understand__. Can you explain it a different way?
• Why did you use__? (for example, a particular strategy or tool)
• How did you come up with__? (for example, an idea or strategy)

Once all students have answered the ASK question, have volunteers articulate their thinking to the whole class so other students can benefit from hearing their strategies.” While these generic question and sentence stems are provided, there is no further guidance or examples for how or when they should be used.

The majority of opportunities for students to engage in  MP3 occur in the extension problems. These include sample answers and often refer teachers back to the prompts listed on page A-43, but give no further guidance on how to build students ability to construct an argument around their thinking or how to critique the reasoning of others. Teachers are often prompted, “In pairs, have students explain their thinking. Do they agree with each other? Discuss why or why not”, however, no guidance is given as to which questions to ask in regards to that specific problem, how to help the students defend their answer, or why an answer makes sense. Additionally, materials include some sample explanations relating to the correct answer being given, but do not always give guidance for teachers on how to effectively guide the conversation if an incorrect answer is being defended. Examples include:

• Teacher Resource, Part 1, Unit 8, Lesson MD2-12, Extension 1, “a) Tom measured the lines to the closest centimeter. (Picture shows two lines above a ruler, one a little shorter than 6cm, one a little longer than 6 cm.) He says that the line on top is about 6 cm long and that the line on the bottom is about 6 cm long. Is that correct? b) Tom subtracts the lengths of the lines in part a): 6 cm - 6 cm = 0 cm. He says that the lines are the same length. Is that correct? Explain. c) In pairs, share your explanations to part b). Do you agree with each other? Discuss why or why not.” Selected sample answers are included. The text then states, “NOTE: For part c), encourage partners to ask questions to understand and challenge each other’s thinking (MP.3) and use of math words (MP.6)-see page A-43 for sample sentence and question stems to guide students.” No further guidance is given to the teacher to help students construct an argument and defend their answer, or which sentence stems and questions from page A-43 should be used specifically for this problem to guide the students’ conversation.
• Teacher Resource, Part 2, Unit 3, Lesson OA2-58, Extension 3, “Raj says that 800 - 573 is the same as 226 + 1. a) Do you agree with Raj? How do you think Raj came up with his addition? Explain. b) In pairs, explain your answer to part a). Do you agree with each other? Discuss why or why not? Sample answer: a) I agree with Raj. To subtract from 800, I can subtract from 799 and add 1 (when I subtract from 799, I don’t need to regroup). 799 - 573 = 226, so 800 - 573 = 226 + 1.” No guidance is given for the teacher to help students with misconceptions about his reasoning that they may have, how to guide students in determining how he came up with his addition, or how to construct an argument to explain and defend their thinking.
• Teacher Resource, Part 2, Unit 8, Lesson G2-2, Extension 2, “Carl says it is possible to draw a closed shape with straight sides and exactly two vertices, because you can extend the two sides as long as you need to. Do you agree with Carl? Explain why or why not? Answer: I do not agree with Carl. If you have two straight sides that meet at one vertex, the straight sides never curve back to close the shape, even if you extend the straight sides.” While an answer is given, no guidance is given for the teacher in guiding the conversation if a student defends an incorrect answer.

The instructional materials reviewed for JUMP Math Grade 2 partially meet expectations for explicitly attending to the specialized language of mathematics.

Accurate mathematics vocabulary is present in the materials; however, while vocabulary is identified throughout the materials, there is no explicit directions for instruction of the vocabulary in the teacher materials of the lesson. Examples include, but are not limited to: 

• Vocabulary is identified in the “Terminology” section at the beginning of each unit.
• “Vocabulary” is identified at the beginning of each lesson.
• The vocabulary words and definitions are bold within the lesson.
• There is not a glossary.
• There is not a place for the students to practice the new vocabulary in the lessons.