4th Grade - Gateway 2

The instructional materials for ORIGO Stepping Stones 2.0 Grade 4 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 limited opportunities to engage with concepts from a number of perspectives, or to independently demonstrate conceptual understanding throughout the grade.

Domain 4.NBT addresses generalizing place value understanding for multi-digit whole numbers and using place value understanding and properties of operations to perform multi-digit arithmetic. Opportunities for students to work with multiplication and division through the use of visual representations and different strategies are included. For example:

• In Module 1, Lesson 2, students write a 5-digit number; then draw a representation of that number on an abacus. Students are led in a discussion of what adding a bead to 90,000 in the ten-thousands place would do and are led to discuss that this would result in a number that is 10 times 10,000 (4.NBT.1).
• In Module 2, Investigation 1, students come up with different strategies to add “537 + 374” (4.NBT.4).
• In Module 6, Lesson 2, introduces a partial product strategy for students to use to solve 3-digit by 1-digit multiplication. Student partition the three-digit factor using place value to multiply (4.NBT.5).

Cluster 4.NF.A addresses extending understanding of fraction equivalence and ordering, building fractions from unit fractions by applying and extending previous understandings of operations on whole numbers and understanding decimal notation for fractions, and compare decimal fractions. Multiple modules present a variety of problems using 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 4, Lessons 12, students begin the lesson by making three jumps on a number line between 0 and 2, deciding how long each jump is, then writing an equation to represent the three jumps. For example, an equation might be 1/2+1/2 +1/2=1 1/2 . During the Step Up portion of the lesson, students use shaded parts that represent mixed numbers to “write the equivalent mixed number and common fraction” represented by the shaded parts (4.NF.1).
• In Module 9, Lesson 2, students discuss unit fractions as fractions with a numerator of 1 and unit fractions with common denominators are added to make other fractions such as adding 1/4+1/4+1/4 to get 3/4 . In the student materials, fractions are represented by fractions strips where the top strip equals 1 whole and each subsequent strip is divided into unit fractions of 1/2, 1/3, 1/4, 1/5. Students discuss which fraction is greater and tell how they decide. In the Step Up portion of the lesson, students are presented with two fraction strips side by side and compare fractions such as 10/6 to 10/12 (4.NF.2).

The instructional materials present 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 1, Lesson 6, Number: Locating 6-digit numbers on a number line, students are shown how to plot large numbers on a number line. For all problems in the Student Journal, number lines are provided with correct spacing and students are either placing numbers on the number line or stating which number is being represented on the number line. Placing numbers on a number line does not help students develop the understanding that a digit in the tens place is ten times larger than the digit to its right nor does it help students with their conceptual understanding of rounding (4.NBT.A).
• In Module 1, Lesson 2, students have little opportunity to build conceptual understanding with the number 326,417. Students are to write it in expanded form and work in pairs to record the different addition expressions (4.NBT.A).
• In Module 2, Investigation 1, students determine different written methods to calculate 537 + 374 and record their results to show each of the methods. Enrichment activities provide some opportunity to understand mathematical concepts through exploring ways to spend $750 (4.NBT.B).
• In Module 7, Lesson 2, students are shown boxes with arrows to assist them in halving numbers, then given base ten blocks and a bill for $92 and told to use the blocks to determine how much each of the four friends will pay. Students then find the answer of 72 divided by 4 by finding half of 72. In the Student Journal students solve division problems using boxes to half enough times to calculate the answer when dividing by four or eight. It is never suggested that students could use other methods such as rectangular arrays or area models to demonstrate conceptual understanding of division (4.NBT.2).

The instructional materials for ORIGO Stepping Stones 2.0 Grade 4 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:, students practice procedural skill and fluency using an area model to multiply whole numbers.

• In Module 2, Lesson 9 (4.NBT.), students use an array model to reinforce the relationship between pairs of tens and fives facts.
• In Module 8, Lesson 9, 4.NBT.5, students practice procedural skill and fluency using an area model to multiply whole numbers.
• Each module contains one lesson Solving Word Problems which requires application. For example, in Module 10, Lesson 12, students must solve a word problem and show their thinking. Question 1b states, "Vishaya rides 4.6 miles. Jacob rides 8.3 miles more than Vishaya. How far does Jacob ride?"

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

• In Module 2, Lesson 5, Student Journal Activity, students use the standard algorithm to identify steps already completed in a problem, write out the numbers, and explain what the number “11” represents in the problem.
• In Module 9, Problem Solving Activity 1, “Rita baked 2 cakes that were exactly the same size for a party. The first cake was a carrot cake which was cut into 8 equal slices. The second cake was a fruit cake which was cut into 12 equal slices. At the party, 3/8 of the carrot cake was eaten and 6 slices of the fruit cake was eaten. Was more of the carrot cake or fruit cake eaten?” Students need to apply their conceptual understanding of fractions.
• In Module 6, Thinking Task, students use information from the chart provided to solve and answer the questions, “What group can take the van? Fill in the total miles walked on the table above. Write which group can take the van. Show your thinking.”

The instructional materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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 5, Lesson 3, students are told to use tape diagrams to model the mathematics.
• In Module 6, Lesson 3, students use arrays to model partitioning strategies for multiplication. Students do not have opportunities to connect the arrays to different representations.
• In Module 10, Lesson 8, students are given models to compare decimals. A support page is provided for the students with number lines, area models, and place value “expanders” for students to use to compare decimals. Students do not have the opportunity to choose or come with their own model and strategies to compare decimals.

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 3, Lesson 9, students use tiles and rulers to determine the area of rectangles. Students are directed which tool they should use.
• In Module 7, Lesson 8, students choose a calculation method but are told which tools to use from the online tool “Flare” with access to number line cards, to find the difference between two numbers.
• In Module 9, Lesson 7, students are locating decimal fractions to the tenths and hundredths on a number line. Students are provided number lines which are already numbered on which to place the decimals.

The instructional materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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 is not 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 10, vocabulary includes words such as decimal, decimal fractions, hundredths, and tenths.
• 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 7, Lesson 6, mixed number is present in the Student Journal, but the definition is not introduced in any lesson in Module 7.