4th Grade - Gateway 1

The materials reviewed for Math Mammoth Grade 4, Light Blue Series, do not meet expectations for assessing grade-level content and, if applicable, content from earlier grades. The curriculum has a Grade 4 Tests and Cumulative Reviews section which includes an End-of-Chapter test for each chapter, a Cumulative Review for every chapter after Chapter 1 and one End-of-the-Year test.  The assessments include an item that addresses and explicitly assess topics from the following: (i) Probability, including chance, likely outcomes, and/or probability models, (ii) Statistical distributions, including center, variation, clumping, outliers, mean, median, mode, range, and/or quartiles; and statistical association or trends, including two-way tables, bivariate measurement data, scatter plots, trend line, line of best fit, and/or correlation, and/or (iii) Similarity, transformations, and/or congruence. 

Note: The user guide identifies the tests as optional; however, the publisher orientation session identified these assessments as a required component for school use. Therefore, the tests are included in this review.

Example of an assessment item that addresses the following items (i) probability; (ii) statistical distributions; (iii) similarity, transformations, and/or congruence:

• Tests and Cumulative Reviews, Chapter 5 Test, Question 8, “Ashley found four different pairs of dress shoes in a store, with prices of $$\$39$$, $$\$45$$, $$\$63$$, and $$\$41$$.  What is their average price?” (6.SP.3)

The materials reviewed for Math Mammoth Grade 4, Light Blue Series, do not meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. The materials provide limited opportunities for all students to engage in extensive work and/or do not provide opportunities for students to meet the full intent with grade-level-problems for standards including but not limited to: 4.OA.4, 4.OA.5, 4.NBT.1, 4.NBT.6 and  4.NF.5 . Some off-grade-level work negatively impacts students’ work with grade-level content..

The materials are divided into two Worktexts, 4-A and 4-B. Each Worktext is divided into chapters. Each chapter is divided into lessons that contain content instruction, mental math problems, puzzle corners, and practice problems, in addition to chapter reviews and a chapter test.

Examples of extensive work with grade-level problems to meet the full intent of some grade-level standards include:

• Worktext 4-B, Chapter 6: Geometry, Review: Area of Rectangles, Question 7, “This is a plan for a two-part clubhouse. a. Write a multiplication and addition sentence for the total area, thinking of one rectangle or two. Solve it.” In Problem Solving: Area of Rectangles, Question 4, “The picture shows a rectangular piece of land with a house in the middle. The plot of land measures 48 ft by 48 ft and the house measures 30 ft by 20 ft.  What is the area of the lawn?” In Review: Area and Perimeter, Question 4, “Solve. Write a number sentence with an unknown for problems a, b, and c. d. A square has an area of 25 cm^2.. How long is its side? This time, don’t worry about writing a number sentence, but you may, if you are able. What is its perimeter?” Students engage in extensive work with grade-level problems to meet the full intent of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real world and mathematical problems.)

• Worktext 4-B, Chapter 6: Geometry, Parallel and Perpendicular Lines, Question 1, “Do these lines intersect or are they parallel? Continue the lines with your ruler.” Students are given a picture of two lines. Question 4, “Draw a line that is perpendicular to the given line and goes throught the given point.” Students are given a picture of a line with a point at the center of it. Question 10, “Find rays, lines,and line segments that are either parallel or perpendicular to each other.” Three different geometric figures are given, and students are tasked with finding different things for each geometric figure. For example, for figure a. students must find two sets of perpendicular lines and one set of parallel lines. Students engage in extensive work with grade-level problems to meet the full intent of 4.G.1 (Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.)

• Worktext 4-B, Chapter 7: Fractions, Equivalent Fractions, Question 1, “Color the first fraction. Shade the same amount of pie in the second picture. Write the second fraction. a. \frac{1}{2}= ____” Students are given a picture of two circles between an equal sign, one circle is divided in half and the other is divided into eights. Question 3, “Mark the equivalent fractions on the number lines. \frac{4}{5} = \frac{a}{b}” Two number lines are given, one is marked in fifths and the other tenths. Question 4, “Split both the colored and white pieces as instructed. Write the fraction after you change it. d. Split all the pieces into three new ones. \frac{1}{3} = \frac{a}{b}” Students are provided with a bar divided into thirds one of the thirds is colored green. Students engage in extensive work with grade-level problems to meet the full intent of 4.NF.1 (Explain why a fraction a/b is equivalent to a fraction (n x a )/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.)

The materials provide limited opportunities for all students to engage in extensive work with grade-level problems and/or do not provide opportunities for students to meet the full intent of most of the standards. Examples include, but are not limited to: 

• Students do not have the opportunity to engage in extensive work with 4.OA.4 (Find all factor pairs for a whole number in the range 1-100.  Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.). In Worktext 4-B, Chapter 5: Division, Divisibility, Question 5, “a. Make a list of multiples of 11, starting at 0 and continue at least to 154. b. Make a list of multiples of 111, starting at 0. Continue as long as you can in this space!” Students are not provided with extensive work that a whole number is a multiple of each of its factors. 

• Students are not provided the opportunity to engage with the full intent of 4.NBT.1 (Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.). In Worktext 4-A, Chapter 2: Large Numbers and Place Value, Multiples of 10, 100, and 1000, Question 1, “Multiply by 10. a. 11 x 10 = ____ , 29 x 10 = ____” The material states, “Remember the shortcut? To multiply any number by ten, write the number and tag a zero to it.” Throughout, the material focuses on the number system as a base ten system and sufficiently addresses place value as a way to understand numbers, but does not specifically address that each number is ten times the number to its right. 

• Students are not provided the opportunity to engage with the full intent of 4.NBT.6 (Find the whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.). In Worktext 4-B, Chapter 5: Division, Long Division 1, Question 1, “Make groups. Divide. Write the dividend inside the ‘corner’ if it is missing. a. Make 2 groups 2⟌62  b. Make 3 groups 3⟌ ”  For part a students are provide a picture of six ten sticks and two ones cubes, and for part b students are provide a picture of six ten sticks and three ones cubes. These questions use modeling to divide ones cubes, tens sticks, and hundred flats into groups illustrating the process of dividing based on place value. There are no problems using rectangular arrays or area models to divide. 

• Students are not provided the opportunity to engage with the full intent of 4.NBT.6 (Find the whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.). In Worktext 4-B, Chapter 5: Division, Long Division 1, Question 1, “Make groups. Divide. Write the dividend inside the ‘corner’ if it is missing. a. Make 2 groups 2⟌62  b. Make 3 groups 3⟌ ”  For part a students are provide a picture of six ten sticks and two ones cubes, and for part b students are provide a picture of six ten sticks and three ones cubes. These questions use modeling to divide ones cubes, tens sticks, and hundred flats into groups illustrating the process of dividing based on place value. There are no problems using rectangular arrays or area models to divide. 

• Students do not have the opportunity to engage in extensive work with 4.NF.5 (Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.) In Worktext 4-B, Chapter 7: Fractions, Equivalent Fractions, Question 11, “Add a. \frac{1}{10} + \frac{8}{100} \rarr \frac{a}{100} + \frac{8}{100} = ” Students are not provided extensive work with expressing a fraction with denominator 10 as an equivalent fraction with denominator 100, and using this technique to add two fractions with respective denominators 10 and 100.

The materials include some off-grade-level content that negatively impacts students’ work with the grade-level standards. Examples include, but are not limited to: 

• Worktext 4-A, Chapter 3: Multi-Digit Multiplication, Multiply in Columns - the Easy Way, Part 2, Question 4, “a. Multiply $1.83 x 3.” The material introduces decimal multiplication, which does not align to a fourth grade standard.

• Worktext 4-A, Chapter 3: Multi-Digit Multiplication, Order of Operations Again, Question 4, “Solve. c. 10 x 7 x (50 + 30) + 200 = ____” The material introduces the order of operations, which does not align to a fourth grade standard.

• Worktext 4-B, Chapter 5: Division, Long Division with Money, Question 1, “Divide and check with multiplication. a. $$\$25.41$$ ÷ 3 b. $$\$14.88$$ ÷ 4”. The material introduces the standard algorithm for dividing decimals, which does not align to a fourth grade standard.

The materials reviewed for Math Mammoth Grade 4, Light Blue Series, do not meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade. The materials do not devote at least 65$$\%$$ of instructional time to the major clusters of the grade: 

• The number of chapters devoted to major work of the grade (including assessments and supporting work connected to the major work) is 3 out of 8, approximately 38$$\%$$.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 67 out of 133, approximately 50$$\%$$. 

• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 84 out of 176, approximately 48$$\%$$. 

A day-level analysis is most representative of the instructional materials as the lessons typically cover multiple days that focus on major work of the grade. As a result, approximately 48$$\%$$ of the instructional materials focus on major work of the grade.

The materials reviewed for Math Mammoth Grade 4, Light Blue Series, do not meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. While some references to future or earlier grade work does occur in the introduction lesson, these references are limited, and are not always related to grade-level concepts or work. The materials does include a Common Core Alignment Document that lists the grade-level standards addressed in each lesson, however, the document does not include information regarding the progression of the lesson standards between grade-level bands. 

There are some examples of references to future grade content, however these references are not always identified and/or related to grade-level work. Examples include, but are not limited to: 

• Worktext 4-A, Chapter 3: Multi-Digit Multiplication, Introduction, “The lesson So Many of the Same Thing has to do with proportional reasoning. The idea is really simple, and prepares students for learning ratios and proportions in middle school.”

• Worktext 4-A, Chapter 4: Time and Measuring, Introduction, “While the Common Core standards do not include them for fourth grade, I have also included some problems where we convert from a smaller unit to a bigger unit (such as 4,500 ml into 4 L 500 ml or 12 feet into 4 yards) because I feel most children are capable of doing these in fourth grade.”

• Worktext 4-B, Chapter 6: Geometry, Introduction, “We also study triangles and classify them according to their angles (acute, obtuse, or right triangles). Classifying triangles according to their sides (equilateral, isosceles, or scalene) will be studied in 5th grade.”

There are some examples of references to prior grade learning, however not all references relate grade-level concepts explicitly to prior knowledge from earlier grades. Examples include, but are not limited to:

• Worktext 4-B, Chapter 6: Geometry, Introduction, “We start our study of geometry by reviewing the third grade concepts of area and the perimeter of rectangles. Students also apply these concepts in various problems, including problems where they write simple equations and a problem where they explore all possible perimeters for a given area… The focus of the chapter is angles. Students learn about lines, rays, and angles; and about acute, right, obtuse, and straight angles.”

• Worktext 4-B, Chapter 7: Fractions, Introduction, “In third grade, students have studied equivalent fractions and compared some easy fractions. In fourth grade, it is time to expand their knowledge of fraction topics. We study: • mixed numbers • adding and subtracting like fractions and mixed numbers with like fractional parts (sums where the denominators are the same, such as 5/6 + 3/6 or 1 2/3 + 2 1/3) • equivalent fractions (for example, 2/3 = 8/12) • comparing fractions • multiplying a fraction by a whole number (for example 5 × ½)”