5th Grade - Gateway 2

The instructional materials for ORIGO Stepping Stones 2.0 Grade 5 partially meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include some problems and questions that develop conceptual understanding throughout the grade-level. Students have little access to concepts from a number of perspectives or to independently demonstrate conceptual understanding throughout the grade.

Domain 5.NBT addresses understanding the place value system and performing operations with multi-digit whole numbers and with decimals to hundredths. Multiple modules explore a variety of real-world applications using a few mathematical representations. Opportunities exist for students to work with place value that call for conceptual understanding and include the use of some visual representations and different strategies. For example:

• In Module 1, Lesson 5, begins with a discussion of a place value table and students talk about the places that are grouped into three digits such as millions and thousands and how those places make a pattern of hundreds, tens, and ones. Students select random digits and place them onto a place value expander to create a 9-digit number. Students read the number then engage in a discussion that includes saying the tens and ones together in each grouping of numbers (5.NBT.1).
• In Module 3, Lesson 8, comparing and ordering thousandths, the class is divided into two teams and the students play the game “Near or Far”. Each member of the team comes forward, spins the cubes, and creates the largest possible decimal number. Then students place the numbers created in order from greatest to least. In the Student Journal, students place numbers in order using a number line, which is provided, or make comparisons using greater than, less than, or equal. Students do not discuss the conceptual understanding that digits to the left are 10 times larger than digits to the right when making their comparisons (5.NBT.1).

Cluster 5.NF uses equivalent fractions as a strategy to add and subtract fractions. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Multiple modules explore a variety of real-world applications using a few mathematical representations. Opportunities exist for students to work with fractions that call for conceptual understanding and include the use of some visual representations and different strategies. For example:

• In Module 6, Lesson 10, students discuss remainders and the context in which they may be used to answer a questions. For instance, when a remainder of 4 must be divided as units of fertilizer between 6 trees, how could this be done? (5.NF.3).
• In Module 8 Lesson 3, students find a fraction of a whole number with unit fractions using number lines (5.NF.1).

The instructional materials present few opportunities for students to independently demonstrate conceptual understanding throughout the grade-level. In most independent activities students are told how to solve problems. For example:

• In Module 9, Lesson 5, students divide 3 by 1/2 and are shown a pot that is divided into 3 quarts and then divided into 1/2 quarts to see 6 servings. Students then divide 1/2 by 3. They are shown a pot with 1/2 a quart which is then divided into thirds to see each friend will get 1/6 quarts of soup. Students are then shown the same process using bar diagrams. In the Student Journal, students solve five problems using bar diagrams which are provided. Since the diagrams are provided and the questions mirror the example, students do not demonstrate conceptual understanding (5.NF.2).
• In Module 12, Lesson 2, students do not have the opportunity to demonstrate conceptual understanding of division and place value. Students are taught the standard algorithm for division (5.NBT.2).

The instructional materials for ORIGO Stepping Stones 2.0 Grade 5 partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present in the materials, but there is an over-emphasis on procedural skills and fluency.

There is some evidence that the curriculum addresses standards, when called for, with specific and separate aspects of rigor and evidence of opportunities where multiple aspects of rigor are used to support student learning and mastery of the standards. There are multiple lessons where one aspect of rigor is emphasized. The materials have a an emphasis on fluency, procedures and algorithms.

Examples of conceptual understanding, procedural skill and fluency, and application presented separately in the materials include:

• In Module 6, Lesson 6, students explain, “Does it make more sense to write the answer in the format of improper fractions or mixed numbers? Why?”
• In Module, 12 Lesson 2, “Division: Developing the standard algorithm,” students use the standard algorithm for division.
• In Module 12, Problem Solving Activity 4, “Three families are vacationing together. They are equally sharing the hotel cost which is $2,634. Thomas’ family is also renting a car for $348. How much will Thomas’ family have to pay for the car rental and hotel together?”

Examples of students having opportunities to engage in problems that use two or more aspects of rigor, include:

• In Module 7, Lesson 4, Maintaining Concepts and Skills states, “Deon bought a bag of apples that weighed more than 3 kilograms but less than 4 kilograms. Monique bought a bag of apples that weighed 7/8 of a kilogram less than Deon’s bag of apples. Hunter bought the same amount of apples as Deon and Monique together. What could be the mass of the apples each person bought?”
• In Module 2, Problem Solving Activity 2, “Between 1,200 and 1,300 people will attend the Freemont High School graduation. The chairs need to be arranged in a rectangular array and 18 to 23 chairs can fit into a row. How many rows and how many chairs in a row are needed to make sure they have enough chairs for all the people?”
• In Module 3, Thinking Task, Question 3 states, “Nancy stacks a total of 160 blocks to build two towers. Tower A is shaped like a cube. Tower B is shaped like a rectangular-based prism. Write the possible dimensions for each tower. Show your thinking" (two different dimension prisms are provided for students to label with length, width, and height).
• In Module 9, Thinking Task, Question 3 states, “In order to see which type of tree grew the most over the course of one year the club will combine the growth data of each tree measured. For this item: Solve using the order of operations. Compare the total growth of the four types of trees.”

The instructional materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 partially meet expectations that the instructional materials carefully attend to the full meaning of each practice standard. The instructional materials do not attend to the full intent of MP4 and MP5.

For MP4, students are given models to use and have few opportunities to develop their own mathematical models. In addition, students have few opportunities to compare different models in problem contexts. Examples include:

• In Module 1, Lesson 2, students use expanders to help read and write seven-digit numbers.
• In Module 2, Lesson 7, students use base ten blocks to build prisms and explore concepts of volume. Students should evaluate the utility of models to determine which models are most useful and efficient to solve problems, in this lesson students are given the model.
• In Module 4, Lesson 1, students use area models to identify equivalent common fractions. In the Student Journal, an area model is used in the example to show how to find equivalent fractions.

For MP5, students are given few opportunities to use tools strategically, as they are most often given the tools to use for a problem. Examples include:

• In Module 1, Lesson 7, students use the halfway point between two multiples on a number line to help them round.
• In Module 3, Lesson 10, students are rounding decimal fractions to the nearest tenth, hundredth, and thousandth. Students are told to “use the thousandths square on the support page with thousandths models or draw number lines to help their thinking. Some students may prefer place value strategies.” Students are given and told what tools to use to round decimal fractions.
• In Module 5, Lesson 3, students apply the standard algorithm to add decimal fractions with a varying number of places. Students are given place value chart templates and told to estimate before solving. Steps directions states, “Encourage them to first estimate the total for each example, then use their estimate to check the reasonableness of their answer.” Students are given specific tools to add decimals.
• In Module 9, Lesson 3, students calculate division of any whole number by a unit fraction. Instructional Steps state, “Emphasize that when they know how many one-sevenths are in a whole it is only the case of multiplying that number by 8 to determine the number in 8 wholes” (8 ÷ 1/7 ). Students are not given any other way to solve the problem, and the materials state that having students compare their solutions and discuss as a class meets MP5.

The instructional materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 partially meet expectations for explicitly attending to the specialized language of mathematics.

Accurate mathematics vocabulary is present in the materials, but there are no instructions on how to use the language of mathematics. While vocabulary is identified throughout the materials, there are no explicit directions for instruction of the vocabulary for the teacher in the Steps portion of the lesson. Examples include but are not limited to:

• Vocabulary instruction for each module is found under Mathematics, Vocabulary Development. Vocabulary identified in bold print is identified as being developed throughout the module. The targeted module vocabulary words can be printed onto cards under Resources. For example, in Module 12 vocabulary includes words such as division, algorithm, and division bracket.
• Each Module contains a parent newsletter. The newsletter highlights key vocabulary and provides the definition for parents in the Glossary section of the newsletter.
• In Module 6, Lesson 4, common denominator is present in the Student Journal, but the definition is not introduced in any lesson in Module 6.