7th Grade - Gateway 2

The instructional materials reviewed for CK-12 Interactive Middle School Math 7 for CCSS meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade-level.

Chapter 1 has multiple opportunities for students to work independently to develop conceptual understanding of analyzing proportional relationships and using them to solve real-world and mathematical problems (7.RP.A) through the use of interactives. Examples include:

• In Lesson 1.6, Activity 1, students use the interactive to develop understanding of proportional relationships by manipulating numbers to see a total cost based on the amount being bought. The student directions state, “A website offers music downloads for $0.79 per song. Use the slider to see how the cost changes as you increase the number of songs you buy. Use the record button to mark different price points on the table below, then using the data given.” (7.RP.2)
• In Lesson 1.8, Warm-up, students manipulate the sliders in the interactive to solve proportional relationships involving percents by finding out the discount on the amount being spent. The student directions state, “Use the interactive and slide the tape diagram to adjust for each fraction. This will help you determine the discount. Then, subtract the discount from the original price to get the sale price.” (7.RP.3)

Chapter 2 has multiple opportunities for students to work independently to build conceptual understanding of applying and extending previous understandings of operations with fractions (7.NS.A) through the use of interactives. Examples include:

• In Lesson 2.3, Activity 3, students develop a conceptual understanding of the distance between numbers by manipulating the sliders on the interactive activity and answering questions such as, “Which equation models the situation in the problem? A. $$11.15 + (-.67) + (-2.25)$$ B. $$11.15 + .67 + 2.25$$ C. $$11.15 + .67 + (-2.25) $$ D. $$11.15 + (-.67) + 2.25$$”. (7.NS.1)
• In Lesson 2.7, students multiply rational numbers. In Activity 1, the context is owing friends money, and students answer, “Annie owes $6 to 3 friends. How much money does she owe? Remember owing money means you have a negative amount.” In Activity 2, the context is rewinding to the beginning of a TV show. Both of these contexts develop an understanding of multiplying signed rational numbers. (7.NS.2a) Practice questions at the end of the lesson in the student materials include problem 1, $$ (-9) × (+8)$$, and problem 2, $$(-5) ×(3)$$, and practice questions from the teacher materials include problem 1, $$(2)(-8)(-3)$$, and problem 4, $$4 ×(-50)$$.
• In Lesson 2.10, Activity 2, students convert fractions to decimals in the interactive to develop understanding of multiplying and dividing rational numbers. The student directions state, “Use the interactive to match the fractions and decimals in the table. Then, select either T for terminating decimals or R for repeating decimals in the last column.” (7.NS.2)

Chapter 3 has multiple opportunities for students to work independently to build conceptual understanding of using properties of operations to generate equivalent expressions and solving real-life and mathematical problems using numerical and algebraic expressions and equations (7.EE) through the use of interactives. Examples include:

• In Lesson 3.3, Activity 2, students manipulate the interactive to sort expressions that are equivalent to the given expression, which develops their understanding of equivalent expressions. The teacher notes describe how the students will be independently working by stating, “For this interactive, students practice matching equivalent expressions to the expression given at the top. Students can click and drag the expressions on the right into the yes or no column.” (7.EE.2)
• In Lesson 3.7, Activity 1, students develop the conceptual understanding of solving multi-step problems with the interactive by balancing the equations to solve for x. The student directions state, “The interactive will tell you if it is not balanced and when the equation is solved correctly. Click on the buttons at the top of the interactive to add and subtract ones and x's. At the end, division buttons will appear, so that you can isolate x”. (7.EE.3)

The instructional materials reviewed for CK-12 Interactive Middle School Math 7 for CCSS meet expectations for attending to those standards that set an expectation of procedural skill and fluency. The instructional materials develop procedural skill and fluency and provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level, especially where called for by the standards (7.NS.1,2; 7.EE.1,4a).

In Chapter 2, the materials develop and students independently demonstrate procedural skill in adding and subtracting (7.NS.1) and multiplying and dividing (7.NS.2) rational numbers. Examples include:

• In Lesson 2.3, Activity 2 Interactive, students demonstrate procedural skill in adding rational numbers written as decimals. The directions state, “You may remember adding decimals and fractions from last year. Adding decimals is not that much different than adding whole numbers, just make sure you line up the decimal point. As with integers, whichever rational number has the greater absolute value, the answer will have that sign. You may use the interactive below to brush up on adding decimals.” (7.NS.1)
• In Lesson 2.4, the Warm-Up: Subtracting Integers states, “Subtraction is taking away a value from another. Adding -4 would mean moving 4 units to the left. With subtraction it is the opposite. Subtracting -4 would mean moving 4 units to the right. Therefore subtraction can also be defined as adding the opposite. $$2 - (-4) + 2 + 4$$. When doing subtraction problems change the problem to adding the opposite before starting.” Students complete practice problems, for example, Activity 1: Diving Depths, Inline Question 2 states, “If $$-5 - 12$$ models Fatima’s diving depth, what is another way to write this problem?” (7.NS.1)
• In Lesson 2.7, students multiply rational numbers. In Activity 1: Annie’s Debt and Activity 2: TV Show Skip Back, students see the results of multiplying numbers with different signs. In Activity 3: Are you -8?, students determine which expressions are equal to -8. For example, Inline Question 1 states, “How would you multiply $$-\frac{2}{3}×2\frac{3}{4}$$?” The practice questions at the end of the lesson, such as “$$(-5) ×(3)$$,” give independent practice on multiplying integers. In Lesson 2.9, Review Questions, students demonstrate procedural skill in multiplying rational numbers, and some examples include “6. Multiply the following rational numbers. $$\frac{1}{11} ×\frac{22}{21} × \frac{7}{10}$$” and “9. Multiply: $$\frac{1}{3} ×\frac{4}{12} × \frac{2}{9}$$.” (7.NS.2)

In Chapter 3, the materials develop and students independently demonstrate procedural skill in applying properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients (7.EE.1) and writing equations of the form $$px + q = r$$ and $$ p(x + q) = r$$ to solve word problems (7.EE.4a). Examples include:

• In Lesson 3.2, Review Questions, students demonstrate procedural skill in applying properties of operations to expressions with multiple examples. Some examples include, “1. Simplify the expression using the distributive property and combining like terms until there are two terms. $$-5(6t-8)-6(t+3)$$” and “4. Use the distributive property to write an equivalent expression. $$(-x+4)$$.” (7.EE.1)
• In Lesson 3.3, Activity 2: Are You Equivalent?, students develop procedural skill in applying properties of operations to determine equivalent expressions. The materials state, “Analyze the expression $$4(x-3) - 2(5x+6)+10.$$ In the box, there are several other expressions that may or may not be equivalent to it. Sort them depending on if they are equivalent or not to $$4(x-3) - 2(5x+6) +10$$.” Also, students develop skill in the practice questions at the end of the lesson, for example, “8. Simplify the expression $$\frac{4x}{2} - 2(x+13)-5^2$$.” (7.EE.1) 
• In Lesson 3.7, students independently demonstrate procedural skill in solving two-step equations in the Review Questions, for example, “10. $$13 - 8x = -3$$.” (7.EE.4a)

The instructional materials reviewed for CK-12 Interactive Middle School Math 7 for CCSS 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 independently throughout the program materials. Examples include:

• In Lesson 3.1, Activity 2, students develop conceptual understanding of combining like terms. Students sort different parts of expressions in the interactive activity. Teacher directions state, “This interactive is a visual example of combining like terms, given expressions with variables. The instructions mention the blue box on the graph represents 7x. Students will also see four yellow boxes each with their own values. Students can click and drag the red points at the corner of each of the boxes to move the boxes around. Students can add to the blue box by stacking the yellow boxes on top of the blue box ... Students can also visualize subtraction by placing the yellow boxes in the blue box.” (7.EE.2)
• In Lesson 2.4, Interactive 3, students develop procedural skill by practicing subtraction with decimals. For example, Inline question 1 states, “Calculate: -56.902 - 12.45 - (-13.58) - (-27.9).  a) -41.567 b) -16.945 c) 33.124 d) -27.872.”  (7.NS.1)
• In Lesson 1.6, Review Questions, students represent and solve proportional relationships presented through different real-world scenarios. For example, Question 5 states, “The amount of money Sebastian spends on shoes can be represented by the equation y = 50x , where x is the number of pairs of shoes he owns and y is the total cost. How many pairs of shoes does Sebastian own if he's spent $650?” (7.RP.2)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• In Lesson 1.9, Activity 1, students develop understanding of the effects of scale factors in geometric shapes. The materials state, “Take Brainy’s photo and enlarge it and shrink it. See what sort of conclusions you can make about scale drawings. Determine what the scale factor would be if he started with an 8” by 8” photo and made a duplicate of 6” by 6”.” Later in the lesson, students build procedural skill in finding scale factors in the review questions. Review Question 4 states, “A map has a scale of 1 inch = 3 feet. What is the scale factor of the map?”
• In Lesson 4.5, Activity 1: Mark Ups Interactive, students develop a conceptual understanding of using equations for percent problems. The materials state, “Use the interactive below to explore how markup rates affect the sale price of a product. In this interactive, students will get to experiment with markups and item prices, and see how that will affect the resulting purchase price.” Inline Question 3 states, “Change the markup rate to 160%. At this rate, what will you multiply each purchase price by to get the selling price?” In the review questions at the end of the lesson, students apply their knowledge of percents and equations to solving real-world problems. Review Question 2 states, “The marked price of a sweater at the clothing store was $24. During a sale a discount of 25% was given. A further 15% discount was given to the customers who have the store’s credit card. How much would a member customer need to pay for the sweater during the sale if the customer paid with the store's credit card? Round your answer to the nearest cent.”
• In Lesson 6.3, students develop procedural skill in finding the circumference and area of circles. In Activity 2, students “Use 3.14 for ???? to determine the area and circumference of the circles in the interactive.” Inline Question 1 states, “The area of a circle is 81????. What are the steps to find the circumference?” In Activity 4: Room for pi?, students apply their understanding of circles to real-world situations. For example, Inline Question 3 states, “Sherry wants to put some decorative tile around the pool. If each tile is 6 inches long, how would she determine how many tiles she needs?”

The instructional materials reviewed for CK-12 Interactive Middle School Math 7 for CCSS meet expectations for identifying and using the Standards for Mathematical Practice (MPs) to enrich mathematics content within and throughout the grade-level. All MPs are explicitly identified in the Teacher Notes and used to enrich the content. 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.

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

• MP1: Lesson 6.8 states, “In this lesson, students will be working with composite shapes, specifically finding their surface area. Students will use what they know about finding the surface area of rectangular and triangular prisms. Students will learn how to break apart composite solids into smaller shapes they are familiar with. They will need to determine which faces should be included in the total surface area (MP1).”
• MP2: In Lesson 1.3, Activity 1, Interactive states, “This interactive works the same way as the previous one (three sliders and three equal numbers) however, the numerator for the first fraction increases by hundreds up to 4,000, and the denominators increase by ones up to 10. Students can use the questions below to reason quantitatively and contextualize the units within the house painting scenario (MP2).”
• MP4: In Lesson 3.1, Activity 3, Discussion Question states, “Students may think it's silly to compare ordering fast food to combining like terms. However, it is a very useful thing in this scenario. There are several different items to order and it is much easier to order them if it is combined. It is also easier for the restaurant to receive the order in this way because there will be less room for error. Discuss why this is with your students to make the example more applicable (MP4).”
• MP5: Lesson 5.3 states, “It may be helpful for students to try drawing various orientations of one of the triangles onto patty paper or a small piece of tracing paper (MP5). By flipping and rotating the tracing, they can see that the triangle still has the same angle measures and side lengths as the interactive.” 
• MP6: In Lesson 6.2, Activity 3, Discussion Question states, “You just derived the formula for the area of a circle using orange segments! The area of any circle is A=pi r^2. Discuss what all the parts of the formula mean.” This Discussion Question encourages students to use precise mathematical language when discussing what each symbol in the formula represents.
• MP7: In Lesson 3.9, Activity 1, Discussion Question states, “Much like equations, there are several different ways to write equivalent inequalities, in words and in symbols. Both of the phrases in #5 do represent the same inequality, but they will look very different. The first is $$5x + 6 < 9$$ whereas the second is $$9 > 6 + 5x$$. Discuss with students that these are the same (MP7).”
• MP8: In Lesson 5.2, Activity 2 states, “Through the interactive and inline questions, students will use repeated reasoning to develop the general method for finding the range of the length of the third side given two sides (MP8).”

The instructional materials reviewed for CK-12 Interactive Middle School Math 7 for CCSS meet expectations for explicitly attending to the specialized language of mathematics. The materials provide instruction on communicating mathematical thinking using words, diagrams, and symbols. Examples include:

• In Lesson 3.2, Activity 1, Teacher Notes, “For questions #4 and #5, discuss what it means to be a factor. Students commonly get the words “factor and ‘multiple’ confused. It might be hard for students to see that a number with a + sign is a factor. If that’s the case, show them this example; if a = 2, b = 3 and c = 5, then 2 and 8, (b+c), are factors of their product, 16. Notice that 3 nor 5 are factors, but their sum is.”
• In Lesson 5.1, the Teacher’s Edition at the beginning of the lesson includes, “Start by reviewing some important terminology: lines, line segments, types of angles, etc. Some of these terms may be new to students, like complementary and supplementary. If students are having difficulty with all the new vocabulary, you can provide them with a vocabulary toolkit or encourage them to make flash cards.”
• In Lesson 6.2, Warm-Up, “A circle is a 2-dimensional figure such that all of the points are the same distance (the radius) from a fixed point (the center). The radius is a line segment where one endpoint is the center of the circle and the other is on the circle. The diameter of a circle is a line segment with both endpoints on the circle and also passes through the center. The length of the diameter is the same as two radii (plural for radius). In mathematical terms, this would be written d=2r. The distance around a circle is called the circumference. Place the vocabulary terms in the correct spot in the picture below.” 

The materials use precise and accurate terminology and definitions when describing mathematics, and the materials also support students in using the terminology and definitions. There is no separate glossary in these materials, but definitions are found within the units in which the terms are used. The vocabulary words are in bold print. Examples include:

• In Lesson 3.3, Introduction, students read the definition of equivalent expressions, “Two expressions are equivalent if they can be simplified to the same third expression or if one of the expressions can be written like the other. In addition, you can also determine if two expressions are equivalent when values are substituted in for the variable and both arrive at the same answer.” In Activity 1 Interactive, students see an example of identifying equivalent expressions to help them understand the proper use. This is introduced to the students as, “In this interactive, there is a column of expressions and a box with other expressions. Your job is to drag the expressions in the box to its equivalent expression in the column. The first one is done for you.”
• In Lesson 5.1, Activity 1, students read the exact definitions of terms relating to angles: “A line is composed of infinitely many points, but you only need two points to define a line. Three points are used to define an angle, where the middle point is always the vertex.” Students are supported in using the terms to answer Inline Questions where they must identify angle terms from a diagram: “1. (Highlight) Based on the data in the image, select the points collinear with point A. 2. (Drag and Drop) Sort the terms below into the correct categories using the image for reference. Remember multiple elements may use the same points. For example, points C and D could describe both a line segment and a ray. 3. Angles are labeled in the form A B C where the middle letter always describes the vertex. The other two letters may be in either order. Select all the correctly labeled angles below.”
• In Lesson 8.5, Activity 1 includes, “Sampling is the practice of using data obtained from a group to represent a population. A population is a group of objects with a common characteristic. The group selected from the population is called a sample. By studying small groups of a larger population, you can identify trends that might apply to the entire population. Throughout this chapter, you will explore what makes a good sample and how it can be used to make estimates about large populations.”