7th Grade - Gateway 1

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Within the MathLinks: Core 2nd Edition materials, the quizzes and cumulative tests are found online in the Teacher Portal in PDF and editable Microsoft Word versions. Cumulative tests are primarily multiple-choice, while quizzes are typically short answer. Materials assess grade-level standards and do not include above-grade assessment items. Examples include:

• Cumulative Tests, Test 1, Problem 9, students find the probability of a compound event. “If you rolled two number cubes 1,000 times, roughly how many times would you expect to get a sum of 12? Explain.” (7.SP.8)

• Unit 3, Quiz A, Problem 4, students solve real world problems with rational numbers and find a unit rate. “Marcus rode his bike at a rate of 2\frac{1}{2} miles in \frac{1}{6} hour. At this rate, how far would he go in one hour?” (7.NS.3, 7.RP.1)

• Cumulative Tests, Test 6, Problem 4, students apply properties of operations to simplify rational coefficient expressions. “Choose ALL expressions below that are equivalent to 3(-d+1.8x)–4.2d. A) -1.2d+5.4x, B)  5.4x+1.2d, C)  5.4x–7.2d, D)  -7.2d+1.8x.” (7.EE.1)

• Unit 7, Quiz B, Problem 6, students demonstrate understanding of how rewriting an expression in a different form can show how quantities are related. “Explain why P=2l+2w and P = 2(l + w) both can be used to find perimeter (P) of a rectangle where l is length and w is width.” (7.EE.2)

• Unit 9 Task - The American Flag, Problem 2, students find what percent is lost based on area. “The original Star-Spangled Banner measured 30 feet by 42 feet when it was created in 1813. In the 1800s, a few people were given pieces of the original flag as mementoes. Part of the flag was lost due to wear and tear through use. The flag was given to the National Museum of American History in 1912. Today, the flag measures about 30 feet by 34 feet. Problem 2) What percent of the area of the original flag was lost or given away?” (7.G.6, 7.RP.3)

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. 

Materials present all students with extensive work with grade-level problems. Examples include:

• Unit 3, Lesson 3.3, Practice 5, Problem 6 students use their knowledge of finding the unit rate to answer questions. “While waiting for the bus, you notice that 3 trucks drive by for every 10 cars. a) At this rate, about how many trucks would you see if 56 cars drove by? b) If you saw 13 trucks drive by, about how many total vehicles drove by during that time?” (7.RP.2)

•  Unit 6, Lesson 6.1, Paintings on the Wall, Problem 1, students use the four operations to solve real-world problems involving rational numbers. “Donna has a room with a wall that is 12\frac{1}{4} feet wide. She wants to paint four square canvases that are all the same size to hang side-by-side across the wall from left to right, and wants to know what size canvases to buy. She wants \frac{3}{4} feet between each of the four canvases. She wants to leave 1\frac{1}{4} feet between the left edge of the wall and the first canvas and 1\frac{1}{4} feet between the right edge of the wall and the last canvas. 1) Sketch and label Donna’s wall with the four canvases on it. Then find the side length for each square canvas.” (7.NS.3)

• Unit 7, Lesson 7.4, Iesha’s Summer, Problem 2, students construct simple equations and inequalities to solve problems by reasoning about the quantities. “For each problem below, write an inequality, solve it, and graph the solutions. Then explain each answer in the context of the problem. 2) A taxi service charges a $2.25 flat rate in addition to $0.64 per mile. Iesha wants to spend no more than $10 on a ride. How many miles can Iesha travel without exceeding her limit?” (7.EE.4)

Materials present opportunities for all students to meet the full intent of the standard. 

• In both the student and teacher editions, grade-level standards for each unit are listed. If the standard is only partially addressed during the unit, the remainder of the text is struck through then identified in a different unit, making it clear when the full intent has been met. For example: 7.SP.7 - “Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy” is first addressed in Unit 1 including both parts, a and b. This standard is addressed again when Unit 4 revisits part b, while Units 9 and 10 revisit part a. Example problems for 7.SP.7 include:

• Unit 1, Lesson 1.1, A Coin Flip Experiment, Problem 10, “Suppose you flipped a coin 20 times and it landed on tails each time. What is the probability of the next flip landing on tails?” 

• Unit 4, Lesson 4.2, A Zero-Sum Game, Problem 4, “Using a paperclip as a spinner,  find the sum for 20 trials and record. Did the results turn out as you expected? Explain.” 

• Unit 9, Lesson 9.2, Penny Drop Probabilites, Problems 1 and 2,”In the Penny Drop Game, a player drops a penny on a board on the floor. If the penny does not land on the board, the player drops it again. If the penny lands on the board and is at least half way in a white space, the player wins. If not, the player loses.  Figures A, B, and C above represent boards for the Penny Drop Game. All three are squares that have side lengths equal to 1 foot. All the circles within board B have the same diameter length. All the circles within board C have the same diameter length. 1) Predict which board you think provides the greatest chance of winning. 2) Test your prediction by calculating the probabilities of winning and losing for each board. What is your conclusion?”

• Unit 10, Lesson 10.1, Revisiting Probability, Problem 4, “Determine if each situation below describes a theoretical probability or experimental probability situation. a)  The XYZ insurance company determines that a 25-year-old male must pay a higher automobile insurance premium than his 56-year-old mother. b) You have a full deck of shuffled playing cards and predict that you have a 25% chance of drawing a card that is clubs.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

To determine the approximate amount of time spent on major work of the grade, materials were analyzed from three different perspectives; units, lessons, and hours. Lesson reviews, unit reviews, and assessment days are included. In addition, supporting work that connects to major work is included.

• The approximate number of units devoted to major work of the grade is 8.5 out of 10, which is approximately 85%.

• The approximate number of lessons devoted to major work is 27 out of 32, which is approximately 84%. 

• The approximate number of hours devoted to major work of the grade is 121 out of 140, which is approximately 86.4%. One hundred forty hours includes all lessons, reviews, and assessments, but it does not include time indicated for intervention, enrichment, and school obligations as those needs vary. 

A lesson-level analysis is most representative of the instructional materials, because the lessons include major work, supporting work connected to major work, and have the review and assessment embedded. Based on this analysis, approximately 84% of the instructional materials for MathLinks: Core 2nd Edition Grade 7 focus on major work of the grade.

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. Connections between supporting and major work enhance focus on major work.

Connections between supporting and major work enhance focus on major work of the grade. Examples include:

• Unit 8, Lesson 8.1, Using Algebra to Find Angle Measures, Problem 3 connects the supporting work of 7.G.5 (Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure) to the major work of 7.EE.4a (Solve word problems leading to equations of the form px+q=r and p(x+q)=r, where p, q, and r are rational numbers). Given three angles that are supplementary, students set up an equation to solve for a variable. “Use an equation to find the measure of the two angles in this diagram that are represented by variable expressions. The diagram is not to scale. Show your work and check your results.” The three angles have measures of (2p+4)\degree, 72\degree, and (3p-6)\degree.

• Unit 9, Lesson 9.3, Practice 7: Extend Your Thinking, Problem 2 connects the supporting work of 7.G.6 (Solve real-world and mathematical problems involving area, volume and surface area, volume, and surface area…) to the major work of 7.NS.3 (Solve real-world and mathematical problems involving the four operations with rational numbers.) “Cubes of edge length \frac{1}{2}-inch are assembled into a pattern. The first three steps are shown to the right. Find the volume and surface area of the solids in the first five steps. Surface area includes all exposed faces, including the “bottom” of the figure.” The three steps show a single cube, then 3 cubes in an “L”, followed by 5 cubes making a longer “L”.

• Unit 10, Lesson 10.2, Practice 3, Problems 5-7 connect the supporting work of 7.SP.2 (Use data from a random sample to draw inferences about a population with an unknown characteristic of interest…) to the major work of 7.EE.3 (Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers…) and 7.RP.3 (Use proportional relationships to solve multi-step ratio and percent problems.) Students use a random sample of housing prices to find percent error. “5) According to McMath’s Realty, the average house in Texas costs 195,000 and the average house in California costs 445,000. Find each data sample’s error as a percent. 6) The percent error of the CA sample is about __ times greater than the TX sample. 7) Which sample is closer to the actual average house cost in the state? How do you know?”

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

There are connections from supporting work to supporting work and/or from major work to major work throughout the grade-level materials, when appropriate. Examples include:

• In Unit 2, Lesson 2.1, Using Coupons Revisited, Problem 2, connects major work of 7.RP.A (Analyze proportional relationships and use them to solve real-world and mathematical problems.) with major work of 7.NS.A (Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.) Students calculate percents and markdowns. “Howard has the same coupons as Bridget, but is going to use them at LOOMY’s Department Store where he may use all four coupons on the same item. He wants to buy a $1,200 TV. Does the order in which Howard uses his coupons matter? Explain how Howard can use all four coupons to get the cheapest TV using words or numbers.”  

• Unit 6, Lesson 6.4, Rewriting Expressions with Fractions, Problem 11, connects major work of 7.EE.A (Use properties of operations to generate equivalent expressions.) with major work of 7.NS.A (Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.) Students determine equivalent expressions using rational numbers. “Circle all the expressions below that are equivalent to \frac{-4x}{6}-\frac{4}{6}: \frac{-4x-4}{6}; \frac{-4(x+1)}{6}; \frac{-2x}{3}-\frac{2}{3}; \frac{-2(x+1)}{3}.”

• Unit 9, Lesson 9.2, Dart Board Probabilities, connects supporting work of 7.SP.B (Draw informal comparative inferences about two populations.) with supporting work of 7.G.B (Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.) Students find the area of circles to calculate the probability of hitting areas in a target. “The dart board game below is made up of concentric circles, which are circles that have the same center. The smallest circle has a 4-inch diameter. Each successive circle has a radius 2 inches greater than the previous one. For a target board game, you earn 2 points if you land on white and 1 point if you land on gray. You win if you earn more gray points than white points. Is this a fair game? Explain.” The target starts with white in the center with 3 additional alternating rings of gray and white.

• Unit 10, Lesson 10.3, Practice 4, Problems 1-4, connect supporting work of 7.SP.A (Use random sampling to draw inferences about a population.) with supporting work of 7.SP.B (Draw informal comparative inferences about two populations.) Students calculate all the measures of the center of random fish populations from 2 lakes. “In Estimating Fish Populations, suppose that when fish were marked, they were also measured. Here are fish lengths in centimeters from two different random samples from two different lakes: Sample A: 75,  32,  38,  42,  47,  68,  51,  51,  61,  31,  51,  62. Sample B: 49,  45,  51,  49,  63,  56,  51,  48,  52,  42,  51,  52. 2) Rewrite each list in order from least to greatest; 3) Calculate statistics for the two data sets; 4) Compare the measures of center for each sample. What do you notice?”

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 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.

Within the Teacher Edition, General Information, each unit provides information about relevant aspects of the content which involve the progression of mathematics. Additionally, Teacher Notes within some lessons identify when current content is building on prior learning and/or connecting to future concepts. Connections to future content and prior knowledge include:

• Unit 2, Algebra in MathLinks: Grade 7, “Algebra topics primarily appear in the CCSS-M Expressions and Equations and Ratios and Proportional Relationships domains. These areas are the focus of four units in MathLinks: Grade 7, and they extend work introduced in 6th grade.” There is a description of the development of Algebra Topics through Unit 2: Percent and Scale, Unit 3: Proportional Relationships, Unit 6: Expressions and Unit 7: Equations and Inequalities. They describe how Units 4 and 5: Rational Number operations extend the use of cups as a manipulative they learned in grade 6 to represent an unknown. Unit 6 includes equations in slope-intercept form “without formally addressing function, slope, and vertical intercept, which is done in 8th grade.”

• Unit 7, Algebra in MathLinks: Grade 7, “In Unit 6, Expressions, students use a visual context to write numerical and algebraic expressions, paving the way to greater flexibility working with variables and expressions. Equations of the y=mx+b are explored without formally addressing function, slope, and vertical intercept, which is done in 8th grade.”

• Unit 7, About the Equation-Solving Sequence, points out that as equations become more complex, students will recognize the benefits of systematic procedures. Therefore, as they are learning procedures for one-step equations in grade 6 and two-step equations in grade 7, it is reasonable to encourage solving mentally to reinforce mathematics as sense-making and value prior knowledge. The progression of lessons in the unit starts with solving mentally, then “re-introduces a more traditional balance technique from 6th grade,” and finally more complicated manipulation of rational numbers.

• Unit 8, Lesson 2, Lesson Notes S8.2a: Sketching Figures, “Students informally begin to think about whether two or more figures exactly cover one another or if sides of one are a multiple of the other. This sets students up for the 8th grade topics of congruence and similarity.”

• Unit 9, Lesson 3, Lesson Notes S9.3: Volume and Surface Area, “Students extend their work finding volumes of right rectangular prisms from 6th grade to include right prisms with other polygonal bases.” Slide 5, “Students find the surface areas of all three prisms. Because the Pythagorean Theorem is needed to compute a value of the length of KD (an 8th grade standard), its measure is given.”

Teacher Edition, Big Ideas and Connections in each unit identifies the focus concepts of the grade level and draws connections among the content specific to the current unit. “Grade 7 is organized around seven big ideas. This graphic provides a snapshot of the ideas in Unit 6 and their connections to each other.” Below the graphic, a chart listing “Prior Work” and “What’s Ahead”, and “These ideas build on past work and prepare students for the future.” Examples include: 

• Unit 6, Teacher Edition, Big Ideas and Connections, Prior Work, “Perform operations with whole numbers, fractions, and decimals. (Grades 3, 4, 5, 6); Extend the number system to include negatives. (Grade 6); Write and interpret numerical expressions. (Grades 5, 6); Solve one-step equations using non-negative numbers. (Grade 6); Explore input-output relationships. (Grade 6)”

• Unit 6, Teacher Edition, Big Ideas and Connections, What’s Ahead, “Analyze and solve linear equations in one or more variables. (Grade 8, HS); Use algebra skills to explore the world of functions. (Grade 8, HS); Use expressions and equations to create mathematical models. (Grade 8, HS)”