7th Grade - Gateway 2

The 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 use 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 materials reviewed for CK-12 Interactive Middle School Math 7 for CCSS meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. 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. Activity 2 states, “Use 3.14 for \pi to determine the area and circumference of the circles in the interactive.” Inline Question 1 states, “The area of a circle is 81\pi. 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 materials reviewed for CK-12 Interactive Middle School Math 7 for CCSS meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials intentionally identify and develop MP1 in connection with grade-level content by providing opportunities for the students to make sense of problems and persevere in solving them. Examples include: 

• In Lesson 4.4, Finding the Whole Given the Percent, MP1 is intentionally developed as they use an Interactive. During Activity 2: Increase and Decrease, “students are asked to find multiple methods for solving for the whole in an increase/decrease problem” as they use proportional relationships to solve multi-step percent problems. Students are asked to prove their method is equivalent using algebraic skills they have learned. (7.RP.3)

• In Lesson 6.11, Volume of Composite 3D Solids, students intentionally develop MP 1 as they “look for an entry point (for finding volume of composite solids) and consider similar problems which may guide them.” In Activity 1: Step Stool Storage, the Teacher Notes explains, “In this Interactive, students will adjust the dimensions of a step stool to figure out its volume and how much storage it has. Students will start with a small step tool with three red points and dimensions, 8\times6\times5. Once students resize the stool to the correct dimensions, a slider will appear at the top of the screen so students can see how the step stool can be broken up into smaller shapes.” (7.G.6)

The materials intentionally identify and develop MP2 in connection with grade-level content by providing opportunities for the students to reason abstractly and quantitatively. Examples include:

• In Lesson 3.4, Equivalent Expressions Within a Context, students intentionally develop MP 2 throughout the lesson using the Interactives and Inline Questions. Activity 3: Mystery Age, states, “Your math teacher, Miss Nomer, gives you an extra credit problem to figure out her age. Half of her age three years ago is equal to one-third of her age nine years from now. How old is she currently? All the pieces to figure out Miss Nomer’s age are in the box below. Your job is to make two equivalent expressions from the clues above. Set them equal to each other so you can determine her age and get extra credit.” (7.EE.4)

• In Lesson 7.3, Discerning Between Equally Likely and Non-Equally Likely Outcomes, students intentionally develop MP2. For example in Activity 1: Rolling the Dice, students are given this situation: “Sai and his friend Kai are rolling dice to see who gets the higher number. They each roll a six-sided die and whoever gets the higher number wins the round. If they roll the same number, then it is a tie and no one wins the round.” Students then are posed the following Discussion Questions, “Look at your results, is this what you expected? Are both boys equally likely to win? Why or why not?” as they use data to determine whether the simulation was fair or unfair. (7.SP.7)

The materials reviewed for CK-12 Interactive Middle School Math 7 for CCSS meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials intentionally identify and develop MP3 in connection with grade-level content by providing opportunities for the students to construct viable arguments and critique the reasoning of others. Examples include:

• In Lesson 2.9, Multi-Step Multiplication & Division Problems involving Rational Numbers, the materials help students to intentionally develop MP3 to its full intent. In the Discussion Question following Activity 1: Switch the Order, students critique the reasoning of a student and form an argument. “Jack claims that the expressions -8 ÷ 2 ÷(-5) ÷ 2 and -8 ⋅ \frac{1}{2} ⋅ (-\frac{1}{5}) ⋅ \frac{1}{2} are the same, do you agree or disagree? Provide evidence for your argument? Can you apply the Commutative or Associative Properties to one and not the other?” (7.NS.2 & 7.NS.3)

• In Lesson 3.8, Using Two-Step Equations to Solve Problems, Activity 2, the Discussion Question asks, “Looking back at the answers for some of the inline questions, are any of the answers not possible? Could you automatically eliminate any?” The Teacher Notes state, “ The answer is, yes, some of the answers could definitely be eliminated. Ultimately, all students will have to take a standardized test at the end of the year, and it is always a good test-taking technique to learn how to eliminate answers that are not possible. In the case of 2, 3, and 5, seconds cannot be negative, so those "distractors" can automatically be eliminated, thus making the selection choice smaller and a greater likelihood of selecting the correct answer. For similar reasons, you could discuss with the class why some of the equations are incorrect in #1 and #5. For example, 160t cannot be positive in the equation because Alex is falling, meaning that 160 needs to be negative.” (7.EE.4)

• In Lesson 4.4, Finding the Whole Given a Percent, Standards for Mathematical Practice: “MP3: In Activity 1, the students are asked to develop an argument around which approach they prefer to help understand a pie chart. The students are given the opportunity to analyze the arguments of their classmates.“ Activity 1: A Slice of Pie, Discussion Question: “When solving the problem above, Donna found it easier to start with smaller percentages, and Ellie found it easier to start with multiples of 10. Who do you side with and why?  Use evidence or an example to support your answer.” The Teacher Notes state, “Answers may vary. Students may have leaned toward small percentages or multiples of 10 for easy calculation. However, finding smaller percentages may be a more effective strategy if the smaller percentages are factors of larger percentages. Allow students to share their strategies.” (7.RP.3)

• In Lesson 8.2, Visually Comparing Two Data Distributions, Activity 3, the Teacher Notes encourage the teacher to have students discuss mean and median in relation to visual data sets. The Discussion Questions asks, “Based on the data taken, which angle do you believe produced the most solar energy?” The Teacher Notes then state, “Allow students to discuss with a classmate and then share with the class. Encourage them to discuss how the mean, median, MAD and IQR influenced their conclusion. After several students have shared their results with the class, allow the groups to put all of their findings together to determine which angle produced the most solar energy.”(7.SP.3)

The materials reviewed for CK-12 Interactive Middle School Math 7 for CCSS meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials intentionally identify and develop MP4 in connection with grade-level content by providing opportunities for the students to model with mathematics. Examples include:                 

• In Lesson 2.5, Finding the Distance Between Two Numbers, students model the rise and run of slope with mathematics. In Activity 2: Getting around Washington DC, students use an online map of Washington, D.C. set in a grid to investigate distances between various landmarks. The Inline Questions state, “1. How many blocks is the Spy Museum for Union Station?  2. Select the two routes Cheryl could take to get to the Spy Museum from 3rd and C SW.  3. The DMV is located at Half St and K in SW. What would be the coordinates on our map?” The Teacher Notes state, this “Interactive is a real-world application of finding the distance between two points when they are not collinear.” (7.NS.1)

• In Lesson 4.9, Solving Problems Involving Taxes, Commissions, and Fees , Activity 4: Big Tipper Continued, students are encouraged to use a variety of strategies to determine a 20% tip, such as multiplying by \frac{1}{5}, using 10% times 2, etc. Then they model with mathematics.  Students are given a sample meal check and are directed as follows: “Put a tip in the input box. Aim for a tip that is around 20%. Click the ‘Check’ button to check your answer. Click the ‘New Amount’ button to try a new bill.” (7.RP.3)

• In Lesson 7.9, Solving Problems Involving Compound Probability, Activity 1: Deal or No Deal, the students model with mathematics the probability of getting a certain prize in a ‘Deal or No Deal’ type game. It states, “You start with four cases, and in each 'round,' choose one case to remove from the table. The goal is to try to leave the case with $10,000 for last. Play the "Deal or No Deal" game in the Interactive below several times. Review the tree diagram and the probability of each final outcome at the end of the game. Once you have a feel for the probabilities of each prize, answer the Inline Questions that follow. 1) How can you find the probability of winning $10,000? 2) Look at the tree diagram. Which of the following statements is true? 3) Look at the completed tree diagram. Select the true statements.” (7.SP.8)

• In Lesson 8.8, Understanding Sampling Variability, Activity 3: Collecting Data, students collect data and analyze data on reaction time, using the Interactive to model with mathematics. The directions on the Interactive are as follows: “You will be recording the time it takes you to catch a bug on the screen 5 times. Then do so with three other classmates and observe the statistics for your results. Press the buttons to release the bug. Tap the bug to catch it, and stop the timer. Observe the results for each student in the given table.”  Students are directed to, “collect data from yourself and from 4 other classmates. Each classmate should provide 5 samples. Display each sample in a spreadsheet. Find the mean and MAD for each person’s data.” (7.SP.2) 

The materials intentionally identify and develop MP5 in connection with grade-level content by providing opportunities for the students to choose tools strategically. Examples include:

• In Lesson 5.2, Triangle Construction,  students do the following:  “In Activity 1, students choose their own tools to create a conjecture about the restrictions on the side lengths of a triangle. The students can come up with a conjecture and provide examples and counterexamples to support their arguments. The students share the experience of how their choice of tool helped or hurt their conjecture.” Activity 1: Three Sides; Discussion Question states: “Can any three lengths create a triangle? Create a conjecture about the side lengths of a triangle.” Then the Teacher Notes state the following:  “For this question, the students will need physical tools to practice constructing triangles based on a chosen number of sides. Allow the students to choose their own tools to practice constructing triangles. The students can choose rulers, graph paper, string, etc. The students should choose three side lengths and then attempt to construct a triangle using those dimensions. If the sum of any two sides is not greater than the third side, the triangle sides will not connect. The students should work in small groups to maximize the data collection and to compare their tool choices with their peers. At the end of the activity, the students should share their conjectures and discuss how their choice of tools assists or hinders their efforts.” (7.G.2)

• In Lesson 6.2, Area and Circumference of Circles: “ In Activity1, the students explore the challenges of using various tools to measure the circumference of a circle.” Activity 1 states, Parts of a Circle; Discussion Question: “What tool do you think would be best for measuring the circumference of a circle? What challenges would you have measuring the circumference with a circle?” After students discuss the questions with their classmates, the Teacher Notes instruct the teacher to print the circle below and allow the students to use real tools like rulers and measuring tape to make the activity more tangible.  (7.G.4)

• In Lesson 7.10, Using Simulations to Estimate Probabilities of Simple Events states, “ In each Activity, the students must choose the appropriate tools to simulate the scenario presented in the Activity.” For example, in Activity 2: Batting Average; Discussion Questions state: “1. What other tools could you have used to simulate a 0.333 batting average, a 1-in-3 chance of getting a hit.  2. Would the tools you chose for the previous question still apply if the batting average was 0.3000 or 0.320? Which tools would work in these situations.” Activity 3: Make a Simulation states, “Choose an event and create a simulation to model its probability. Choose the appropriate tools to model the probability of the event. It may require research to find the probability of the possible outcomes. State the scenario, the probabilities, the research you did, and the simulation tools you chose.” (7.SP.6 & 7.SP.7)

The materials reviews for CK-12 Interactive Middle School Math 7 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students,  in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The materials intentionally develop MP6 through providing instruction on communicating mathematical thinking using words, diagrams, and symbols. Examples include:

• In Lesson 3.2, Rewriting Expressions Using the Distributive Property, Activity 1, students are asked to expand expressions such as \frac{1}{2}(4a-5). The Teacher Notes state, “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+care factors of their product, 16. Notice that 3 nor 5 are factors, but their sum is.” (7.EE.1)

• In Lesson 5.1, Special Angle Pairs, students learn about supplementary, complementary, adjacent, and vertical angles.  The Teacher Notes at the beginning of the lesson states, “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.” (7.G.5)

• In Lesson 6.3, Solving Problems Involving Circles, Activity 2, students attend to precision as they are asked to, “Use 3.14 for \pi to determine the area and circumference of the circles in the Interactive. Repeat several times until you are comfortable with the two formulas. Remember the 3.14 is an estimation of \pi, but it does enable you to get a numerical answer, and not one in terms of \pi.“ (7.G.4)

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, Identifying Equivalent Expressions,  Introduction, students read the definition of equivalent expressions. The definition is as follows: “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.” (7.EE.1)

• In Lesson 5.1, Special Angle Pairs, Activity 1, students read and apply 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 items 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 ∠ABC, where the middle letter always describes the vertex. The other two letters may be in either order. Select all the correctly labeled angles below.”  (7.G.5)

• In Lesson 8.5, Introduction to Sampling, Activity 1 states, “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.”(7.SP.1)

The materials reviewed for CK-12 Interactive Middle School Math 7 for CCSS meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

The materials intentionally identify and develop MP7 in connection with grade-level content by providing opportunities for the students to look for and make use of structure. Examples include:

• In Lesson 2.4, Subtracting Integers and Other Rational Numbers, Activity 1, students develop MP7 as they use an Interactive to “explore the link between adding a negative and subtracting a positive number.”  The Teacher Notes state, “This interactive shows an application of how negative values are used; in this case, it is how deep a diver dives. Students will see a body of water with a diver, boat and some fish. There are also two number lines; the horizontal line ranges from -20 to 0 and the vertical line ranges 0 feet to 20 feet. The horizontal line has a red point that students can click and drag to make the diver dive. While the student moves the point in the negative direction the father down the diver will go. The arrow on the vertical line will travel down with the diver showing the student how many feet down the diver is.”  Discussion Question: “Why is subtracting a negative integer the same as adding a positive integer?” (7.NS.1)

• In Lesson 2.11, Using Arithmetic Methods to Solve Multi-Step Problems, students intentionally develop MP7 as they “explore how parentheses placement can affect the simplification of an expression.” In Activity 1, students make use of structure as they investigate an equation. “See the equation with blue parentheses. Below is the expression from the left side of the equation.  Depending on which operation is performed first, the value of the expression changes.  For example, if you add 3 + 4 before performing any other operations in the first equation, then the expression equals 37. 1) Move the red points to change which operation is performed first. 2) Move the red point so the second expression equals 18. 3) Move the red point so the third expression equals 1.” (7.NS.2a & 7.NS.3)

• In Lesson 3.9, Writing Two-Step Inequalities, Activity 1, MP7 is intentionally developed as students compare the structure of inequalities to the written expression that matches them.  In this interactive, you will practice matching inequalities to phrases.  Try to notice any keywords that help you match the phrase to the inequality.” In the Inline Questions, students continue to use the structure of inequalities and their expression in words. For example, Question 2: “How could the inequality -3>-2+\frac{1}{2}z be expressed differently than it was in the interactive? a) -3 is less than -2 and half a number. b) -3 is greater than -2 and half a number.  c)  -3 is less than half a number and 2 d) -3 is greater than -2 less than half a number. (7.EE.4)

• In Lesson 6.2, Area and Circumference of Circles, Activity 2, Students use an interactive to determine the revolutions of a bowling ball as it rolls down an alley. Students apply their measurements as they use the ratio of circumference to diameter to help derive the circumference of a circle” guided by  Inline Questions: “1). Use the interactive. Approximately how many inches does the ball travel in on revolution?  2) How can we mathematically find the number of revolutions it takes the ball to reach the pins, 3. The number \pi is defined as the ratio of the circumference, C, to the diameter, d.  For the ball, this is \frac{27}{8.5}\approx 3.14 and 4) How can we rewrite the formula  as a formula to find the circumference? r represents the radius in the formulas.” (7.G.4)

The materials intentionally identify and develop MP8 by providing opportunities for the students to look for and express regularity in repeated reasoning. Examples include:

• In Lesson 1.4, Identifying Proportional Relationships, students intentionally develop MP8 as they “use repeated reasoning to develop a conjecture about the properties of graphs of proportional relationships.” Activity 1’s Discussion question asks, “What do you notice about this graph and the graph from the Driving Away interactive?” (7.RP.2)

• In Lesson 2.3, Adding Integers and Other Rational Numbers, Activity 1: In the Discussion Question,  students are asked to make use of repeated reasoning. After an Interactive practice of adding positive and negative integers, students are asked to, “Think about the three cases of integer addition: when the addends (the two numbers that are being added) are both positive, both negative, or one of each.  How would you approach an addition problem differently in each of these cases and what sign the answer has?” The Teacher Notes states the following: “Students should use the interactive to use repeated reasoning and look for patterns that can be generalized for all integer addition problems.” (7.NS.1)

• In Lesson 6.10, Volume of Triangular Prisms, in the introduction, students compute the area of a triangle using a one square unit grid and then use that result to compute the volume of a one unit high triangular prism.  The Discussion Questions 3 and 4 lead them to use repeated reasoning to derive the formula for the volume of a triangular prism of any dimensions: 3) How many cubes would there be in 2 of these figures? 4) How many cubes would there be in n of these figures?” (7.G.6)