6th Grade - Gateway 2

The instructional materials reviewed for CK-12 Interactive Middle School Math 6 for CCSS partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. The materials include multiple opportunities for students to engage in routine application of grade-level skills and knowledge, within instruction and independently. The materials include one non-routine application problem within instruction, but students do not demonstrate independent application of mathematics in non-routine situations.

Examples of students engaging in routine application of grade-level skills and knowledge, within instruction and independently, include: 

• In Chapters 1 and 2, students engage in familiar real-world scenarios and demonstrate the application of ratio and rate reasoning through the use of questions and interactives (6.RP.3). For example, in Lesson 1.3, Activity 4, Inline Question 1 states, “You mix 12 cups of brown paint by using 3 cups of yellow paint to 4 cups of red paint to 5 cups of blue paint. After painting a portion of a fence you realize that you need more of the same color paint to finish the fence. This time you want to make 26 cups. How many cups of each color do you need?” The answers are multiple choice, and students can use the interactive to check each choice. Also, in Lesson 2.9, Activity 2 states, “1. What do you notice about the relationship between the decimal value and its percent? A. The percent is 10 times the decimal value. B. The percent is 100 times the decimal value. C. The percent is greater than the decimal value.”
• In Chapter 7, students write and solve equations of the form $$x + p = q$$ and $$px = q$$ (6.EE.7) and use variables to represent two quantities in a real-world problem that change in relationship to one another through the use of interactives (6.EE.9). For example, In Lesson 7.3, Activity 1 Interactive states, “The typical lithium element has 3 protons and 7 total protons and neutrons. You can represent this relationship using an equation: Protons + Neutrons = Mass Number, $$P + N = M$$. Substitute our known values to produce the following: $$3 + N = 7$$. Use this equation in the interactive below to visually identify the number of neutrons in the lithium atom.” Also, Lesson 7.8, Activity 1 states, “This interactive gives students the chance to use the formula distance = rate x time. To start, select an animal from the drop down menu. The rate will be given at the top and below the students can fill out the table of values for the distance that the animal travels at certain times. If an incorrect value is entered, it will turn red. Once a correct value is entered it will turn black. Students can try different animals by clicking the reset button under the interactive window.” Inline Questions help to formulate an equation, for example, Inline Question 2 states, “Complete the table for the cheetah, The cheetah travels at 105 feet per second. Write an equation for the cheetah’s distance d over time t.”
• In Lesson 9.4, Activity 2, students apply their knowledge of the area of a triangle in real-world contexts (6.G.1) as they answer, “Marielle wants to paint a triangular section of her house. One gallon of paint covers 400 square feet. Use the interactive below to find the dimensions of the triangle section.” Inline Question 4 includes, “If Marielle wants to use three layers of paint on the triangular section of her house, how many gallons will be needed?” 
• In Lesson 10.5, Activity 3, Inline Question 4, students use mean and median to analyze a set of data as they answer, “Look at Helena’s time again: 17.44, 17.5, 17.85, 17.99, 18.11, 18.25, 31.23, 35.55. Remember, the mean time is 21.74 seconds. The median time is 18.05 seconds. What can you conclude by comparing the mean (average) to median (middle)?”

The non-routine application problem within instruction is in Chapter 3. In Lesson 3.9, Activity 3 Interactive, the teacher's directions include, “In this interactive, students will create their own division of fractions problem and use a visual model to help solve the problem. To begin, students can type in the two fractions they want to divide. After the fractions are entered the tape diagrams will show the fraction represented as a diagram. Students can click and drag the red diagram over the blue diagram, then the division equation will appear. Click the Show divisions button to see how the division is represented on the diagram. Students can then type in their answer to the division statement in the textbox. Students will type in the value of the numerator, then the value for the denominator.” (6.NS.1)

The instructional materials reviewed for CK-12 Interactive Middle School Math 6 for CCSS partially meet expectations for identifying and using the Standards for Mathematical Practice (MPs) to enrich mathematics content within and throughout the grade-level. The materials state that teachers should use a few MPs in each lesson, but each lesson does not include guidance on which MPs to use. MPs are explicitly identified in the Teacher Notes, but MPs 7 and 8 are not identified in the materials. Also, no MPs are identified in Chapters 3 and 10, and they are only identified once in Chapters 6, 7, and 8.

Examples of the materials identifying and using the MPs to enrich the mathematics content include:

• MP1: This was identified two times in the materials. Lesson 4.6 states, “Students will then work through some division problems within a context and work through a long division puzzle (MP1). If students are still not comfortable with long division, they can work through the widgets at the beginning as many times as they wish. For students who do not need to spend as much time practicing, encourage them to try to challenge problems at the end of the lesson.” Also, Lesson 4.9 states, “students will learn about least common multiples. Students will work through real-world examples and puzzles to practice finding the least common multiple of two or more numbers (MP1).”
• MP2: Lesson 1.1 states, “In this lesson, students will learn that a ratio is used to describe the relationship between two quantities.Throughout the lesson it would be helpful to allow them to use the language. This will help them understand how they can use ratio language in everyday situations. With partners, they can compare the number of students, chairs, desks, windows, whiteboards, etc. using ratio language (MP2).”
• MP4: Lesson 7.6 states, “As students write their equations, they should be thinking about the independent and dependent variables to help them understand that the equations represent the relationship between the two variables. Once students have written the equations, they should practice interpreting what each part means within the context of the situation (MP4).”
• MP6: Lesson 2.3 states, “In this lesson, students will review long division and apply it to solving unit pricing problems. Students will be filling out tables, but it is important for them to practice using the language associated with unit rates. Students can practice reviewing their answers with a partner using ‘price per item’ phrases (MP6).”

There are some instances in which the Mathematical Practices are labeled but do not enrich the content. For example, in Lesson 4.4, MP5 is identified. Students use the Interactive to solve the equation, but they can type numbers into the Interactive until it shows the correct answer, which means students are not using tools strategically to enrich their work with dividing decimals using the standard algorithm. In Activity 2, this process is described in the Teacher Notes, “This interactive gives students a walk through for multiplying two decimals. Use the text boxes to evaluate the product one step at a time, after a student has typed in their answer they should press the enter key to see if it is correct. If it is wrong, it will turn red and students can try again. Once a correct answer is entered it will turn black and a new text box will appear. Students can use the text boxes above each number being multiplied to help with their calculation, but it is not necessary. Once students have multiplied and added all the numbers, they will be able to place the decimal by clicking and dragging the red point to one of the red circles in between each of the numbers in the final answer. Once the student has placed the decimal, click the Check button.”

The instructional materials reviewed for CK-12 Interactive Middle School Math 6 for CCSS partially meet expectations for carefully attending to the full meaning of each practice standard. The materials do not attend to the full meaning of two MPs.

Examples of the materials not attending to the full meaning of MP5 include, but are not limited to: 

• In Lesson 4.4, the Introduction includes, “Students will continue practicing multiplying decimals; however, this lesson focused on the standard method. Students will begin working with an interactive similar to the ones found in the previous lesson, Multiplying Decimals with Diagrams. There is a widget interactive that will guide students through the standard method of multiplying decimals (MP5).” Students do not select which tools to use as they are provided.
• In Lesson 5.4, Activity 2, “In the previous activity, we saw that numbers on both sides of the number line are related. The overlapping numbers in the interactive above are opposite numbers. The opposite of a number, also known as the additive inverse, comes from the additive inverse property. The additive inverse property states that for any real number n, there exists an additive inverse which is the same distance from zero in the opposite direction. Additionally, the sum of a number and its additive inverse will always be zero: n + (-n) = 0. A trick is to change the sign of the number, for example, the opposite of 7 is -7 and the opposite of -15 is 15. Use the interactive below to explore opposite numbers.” Students do not select a tool for this investigation.

Examples of the materials not attending to the full meaning of MP8 include, but are not limited to:

• In Lesson 3.9, students use tape diagrams to practice dividing fractions. In the Warm-Up, students answer, “How many times can you reload the water gun?” The materials include, “Use the interactive to see how many times you can reload the water gun without having to run to fill the bucket up with more water. Through this lesson, you will use tape diagrams to model fraction division and solve for the quotients.” Students are given the tape diagram that represents the expression and the steps to solve it. Students do not generalize a pattern from this activity.
• In Lesson 4.4, Activity 1, Inline question 3, students answer, “What pattern do you see when finding the number of Newtons? Remember that you multiply the number of kilograms by 10 to find the Newtons. Red Car: 0.454 kgs, 4.54 Newtons; Green Car: 0.681 kgs, 6.81 Newtons; Blue Car: 1.362 kgs, 13.62 Newtons. a) Multiplying by 10 adds a zero to the end of the number. b) Multiplying by 10 moves the decimal place over to the left by 1 place. c) Multiplying by 10 moves the decimal point over to the right by 2 places. d)Multiplying by 10 moves the decimal point over to the right by 1 place.” Students choose from a finite list, which they can do by guessing, instead of generalizing a pattern from repeated reasoning.

The instructional materials reviewed for CK-12 Interactive Middle School Math 6 for CCSS partially meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The materials provide opportunities for the students to construct arguments about the content, and examples include:

• In Lesson 1.3, Activity 4, Discussion Question, students construct an argument about how the ratio staying the same will impact the results of the paint mixing while taking equvalentance into account. The materials state, “When the total amount of paint increases and decreases but the ratio of paint colors remains equivalent, does the shade of brown change? Use the idea of equivalence to help explain why.”
• In Lesson 2.2, Activity 3, Discussion Question, students construct an argument about the appropriate use of unit rates in relation to shark teeth. The materials state, “Use the unit rates to order the sharks from greatest number of teeth to least number of teeth. Did you need to find the unit rate to order the sharks? If not, think of an example of when you would need to find the unit rate to compare numbers of shark teeth.”
• In Lesson 6.8, Activity 2, Discussion Question, students construct an argument to determine the appropriate time to substitute numbers in an expression. The question states, “You can only sometimes count the numbers of bumps on the spiral. When is it useful to substitute numbers into the expression to find the number of bumps?”

There are no opportunities for students to analyze the arguments of others, and examples include, but are not limited to:

• In Lesson 1.4, Activity 1, Discussion Question, students discuss with the class but do not analyze the arguments of others. The question states, “What are the differences between a tape diagram and a number line? Discuss your answer with your class or in the CK-12 Cafe.” 
• In Lesson 3.6, Activity 2, Discussion Question, students use mathematics to explain their thinking but do not analyze other students’ reasoning. The question states, “The largest IMAX theater in the world is in Melbourne, Australia. It is 105 feet by 75 feet. Compare the area of the largest IMAX screen with the standard IMAX screen and the "local movie theater" screen from the example.”
• In Lesson 10.5, Warm-Up, the Discussion Question states, “What is the difference between center and variability? Discuss in class or in the CK-12 Cafe. Answers may vary. Encourage students to point out that it is not guaranteed the 50% of the data values occur above or below the mean.”
• In Lesson 10.8, Activity 3, Discussion Question, students describe which data set to use but do not analyze the arguments of others. The question states, “The mean of the dataset is 16.92 ounces and the mean absolute deviation is approximately 1.15 ounces. Which set of statistics do you feel better describes the data: the median and the interquartile range, or the mean and the mean absolute deviation (MAD)?”

The instructional materials reviewed for CK-12 Interactive Middle School Math 6 for CCSS partially 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 include some examples of assisting teachers in engaging students to construct viable arguments and analyze the arguments of others, but there are also multiple instances where the materials do not assist teachers.

Examples of the materials assisting teachers to engage students in constructing and/or analyzing the arguments of others include:

• In Lesson 4.8, Activity 1, Interactive Plix: GCF Using Lists, the Teacher Notes state, “The last question prompts students to construct an argument for why the GCF is always a natural number and never a fraction. You can remind students that natural numbers are whole numbers not including the number 0 (1, 2, 3, 4, etc).” 
• In Lesson 5.1, Warm-Up: Sea Level, the Teacher Notes state, “Allow students to turn and talk and build off each other’s ideas in a whole class setting. By the end of the discourse, students should be able to articulate that the sea animals are below and the birds are above sea level, and that sea level represents 0 because it has no elevation, the other animals are located somewhere in relation to sea level.”
• In Lesson 10.9, Activity 3, Discussion Question, the Teacher Notes state, “Answers may vary. Encourage students to use numerical evidence from the datasets to support their argument. One missing piece of information is the reason behind the two grades of 0. Perhaps the students did not know any answers or perhaps they were absent. Knowing this information would likely affect the students' stance on which side did better.”

Examples where the materials do not assist teachers to engage students in constructing and/or analyzing the arguments of others include:

• In Lesson 3.8, Activity 2, Supplemental Questions, the Teacher Notes state, “Why is dividing by a fraction, the same as multiplying by the reciprocal of the fraction? Students will have different explanations. They may refer to the tape diagram and explain how many times a fraction can divide into a whole number. They may note that the number of parts the bars are divided into is the denominator and the groupings of the parts is the numerator.” The materials do not assist teachers in having students analyze the arguments of others.
• In Lesson 5.3, Warm-Up, the Teacher Notes encourage students to discuss their answers to questions provided to the teachers. The Teacher Notes state, “Allow students to discuss and answer the questions. How does the use of a negative affect the meaning of the numbers above? The negative identifies whether the money is owned or owed. How does the distance from zero affect the meaning of the numbers above? The closer to zero, the smaller the debt or value. The farther from zero, the greater the debt or value.” The materials encourage discussion among students, but they do not assist teachers in having students analyze the arguments of others.
• In Lesson 8.4, Warm-Up, Discussion Question, the Teacher Notes indicate that there may be different answers from students, but there is no assistance as to how students construct an argument about the weight. The Teacher Notes state, “Allow students to answer. Answers may vary. The beam must weigh either 18 tons or less. The beam could weigh 2 tons. The beam cannot weigh 20 tons.”