7th Grade - Gateway 1

The materials reviewed for Math Nation Grade 7 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

The curriculum consists of nine units, including one optional unit. Assessments include Cool-down Tasks, Mid-Unit Assessments, and End-of-Unit Assessments. Examples of assessment items aligned to grade-level standards include:

• Unit 2, Lesson 1, Cool-down, “Here are three different recipes for Orangey-Pineapple Juice. Two of these mixtures taste the same and one tastes different.  Recipe 1: Mix 4 cups of orange juice with 6 cups of pineapple juice. Recipe 2: Mix 6 cups of orange juice with 9 cups of pineapple juice. Recipe 3: Mix 9 cups of orange juice with 12 cups of pineapple juice. Which two recipes will taste the same, and which one will taste different? Explain or show your reasoning.” (7.RP.2)

• Unit 3, End-of-Unit Assessment (A), Question 2, “The shape is composed of three squares and two semicircles. Select all the expressions that correctly calculate the perimeter of the shape. A. 40+20𝞹; B. 80+20𝞹; C. 120+20𝞹; D. 300+100𝞹; E.10+10+10𝞹+10+10+10𝞹 .” (7.G.4)

• Unit 6, Mid-Unit Assessment (A), Question 3, “At practice, Diego does twice as many push-ups as Noah, and also 40 jumping jacks. He does 62 exercises in total. The equation 2x + 40 = 62 describes this situation. What does the variable x represent? A. The number of jumping jacks Diego does B. The number of push-ups Diego does C. The number of jumping jacks Noah does D. The number of push-ups Noah does” (7.EE.4)

The materials reviewed for Math Nation Grade 7 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials present opportunities for students to engage in extensive work and the full intent of most Grade 7 standards. Each lesson contains a Warm-Up, a minimum of one Exploration Activity, a Lesson Summary, Practice Problems, and three Check Your Understanding Questions. Each unit provides a Readiness Check and a Test Yourself! practice tool.  Examples of full intent include:

• Unit 1, Lesson 7, 1.7.2 Exploration Activity, engages students with the full intent of 7.G.1 (Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale). Students use a diagram to complete a table of values involving scale drawings to compute actual dimensions. “Your teacher will give you a scale drawing of a basketball court. The drawing does not have any measurements labeled, but it says that 1 centimeter represents 2 meters. 1. Measure the distances on the scale drawing that are labeled a-d to the nearest tenth of a centimeter. Record your results in the first row of the table. 2. The statement ‘1 cm represents 2 m’ is the scale of the drawing. It can also be expressed as ‘1 cm to 2 m,’ or ‘1 cm for every 2 m.’ What do you think the scale tells us? 3. How long would each measurement from the first question be on an actual basketball court? Explain or show your reasoning. 4. On an actual basketball court, the bench area is typically 9 meters long. a. Without measuring, determine how long the bench area should be on the scale drawing. b. Check your answer by measuring the bench area on the scale drawing. Did your prediction match your measurement?” Students are provided a table with two rows labeled, “scale drawing” and “actual court,” and four rows labeled “(a) length of court,” “(b) width of court,” “(c) hoop to hoop,” and “(d) 3 point line to sideline,” respectively.

• Unit 5, Lesson 13, 5.13.3, Exploration Activity, engages students with the full intent of 7.NS.A (Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers). Students complete a table of values using the four operations and fractional values to add, subtract, multiply, and divide rational numbers.  For example, “For each set of values for a and b, evaluate the given expressions and record your answers in the table.”

•  Unit 6, Lesson 13, 6.13.2 Exploration Activity, engages students with the full intent of 7.EE.4b (Solve word problems leading to inequalities of the form px+q>r or px+q, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. Students analyze an inequality, explain their reasoning, and graph the results. “A sign next to a roller coaster at an amusement park says, ‘You must be at least 60 inches tall to ride.’ Noah is happy to know that he is tall enough to ride. 1. Noah is x inches tall. Which of the following can be true: x>60, x=60, or x<60? Explain how you know. 2. Noah's friend is 2 inches shorter than Noah. Can you tell if Noah's friend is tall enough to go on the ride? Explain or show your reasoning. 3. List one possible height for Noah that means that his friend is tall enough to go on the ride, and another that means that his friend is too short for the ride. 4. On the number line below, show all the possible heights that Noah's friend could be.”

The materials present opportunities for students to engage with extensive work with grade-level problems. Examples of extensive work include:

• Unit 5, Lesson 14, Exploration Activity, Check Your Understanding, and Practice Problems, engage students in extensive work with 7.RP.2 (Recognize and represent proportional relationships between quantities). In 5.14.2 Exploration Activity, students use proportional knowledge to figure out how long it takes a tank to drain. “A tank of water is being drained. Due to a problem, the sensor does not start working until some time into the draining process. The sensor starts its recording at time zero when there are 770 liters in the tank. 1. Given that the drain empties the tank at a constant rate of 14 liters per minute, complete the table: 2. Later, someone wants to use the data to find out how long the tank had been draining before the sensor started. Complete this table: 3. If the sensor started working 15 minutes into the tank draining, how much was in the tank to begin with?” In 5.14.6 Practice Problem, Question 3, students use proportional knowledge to figure out how much water is in a tank after five minutes. “A large aquarium of water is being filled with a hose. Due to a problem, the sensor does not start working until some time into the filling process. The sensor starts its recording at the time zero minutes. The sensor initially detects the tank has 225 liters of water in it. A. The hose fills the aquarium at a constant rate of 15 liters per minute. What will the sensor read at the time 5 minutes? B. Later, someone wants to use the data to find the amount of water at times before the sensor started. What should the sensor have read at the time −7 minutes?” In 5.14.7 Check Your Understanding, Question 2, students work with proportional knowledge to calculate the total mileage of a runner. “Caroline is training for a half-marathon race and ran 10.1 miles on Monday, 4.3 miles on Tuesday, and 9.9 miles on Thursday. Her friend Xander is also training and he ran \frac{4}{5} of Caroline’s total mileage. How many miles did Xander run? Express your answer as a decimal.”

• Unit 7, Lesson 13, Lesson 15, and Lesson 16, engages students in extensive work with 7.G.6 (Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms). Lesson 13, 7.13.2 Exploration Activity, students solve problems in context using three-dimensional objects. “A box of chocolates is a prism with a base in the shape of a heart and a height of 2 inches. Here are the measurements of the base (picture given). To calculate the volume of the box, three different students each have drawn line segments showing how they plan on finding the area of the heart-shaped box (picture given) students use three-dimensional objects to calculate volume. 1. For each student’s plan, describe the shapes the student must find the area of and the operations they must use to calculate the total area. 2. Although all three methods could work, one of them requires measurements that are not provided. Which one is it?” Lesson 15, 7.15.5 Exploration Activity, students calculate the volume required to fill an object and the area that is left over once the contents are removed. “A wheelbarrow is being used to carry wet concrete. (Picture with dimensions are given) 1. What volume of concrete would it take to fill the tray? 2. After dumping the wet concrete, you notice that a thin film is left on the inside of the tray. What is the area of the concrete coating the tray? (Remember, there is no top.)” Lesson 16, 7.16.2 Exploration Activity, in this activity students work in pairs to solve a real-world problem that involves finding the volume and surface area of a play structure by decomposing the structure into prisms. “At a daycare, Kiran sees children climbing on this foam play structure. (Picture with dimensions are given) Kiran is thinking about building a structure like this for his younger cousins to play on. 1. The entire structure is made out of soft foam so the children don’t hurt themselves. How much foam would Kiran need to build this play structure?  2. The entire structure is covered with vinyl so it is easy to wipe clean. How much vinyl would Kiran need to build this play structure? 3. The foam costs 0.8¢ per in³. Here is a table that lists the costs for different amounts of vinyl. What is the total cost for all the foam and vinyl needed to build this play structure?”

• Unit 8, Lesson 1, Warm-Up, Practice Problems, and Check Your Understanding, engage students in extensive work with 7.SP.6 (Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability). In 8.1.1 Warm-Up, students find the probability of an event to make predictions. “Andre and his dad have been fishing for 2 hours. In that time, they have caught 0 bluegills and 1 yellow perch. The next time Andre gets a bite, what kind of fish do you think it will be? Explain your reasoning.” In 8.1.5 Practice Problems, Question 1, students explain their reasoning in approximating the probability of a chance event. “Lin is interested in how many of her classmates watch her favorite TV show, so she starts asking around at lunch. She gets the following responses: [yes, no], [yes, no], [yes, no], [no, yes], [no, no], [no, no], [no, no]. If she asks one more person randomly in the cafeteria, do you think they will say "yes" or "no"? Explain your reasoning.” In 8.1.6 Check Your Understanding, Question 2, students use probability to determine the probability in students preference in selecting a dog or a cat. “Sam asked some of his classmates if they preferred cats or dogs. Table shows the following responses: [cat, dog],  [dog, cat], [dog, cat], [cat, dog], [dog, cat], [dog, cat]. If Sam continues to ask his classmates, which statement is true based on the table? (A) The next person he asks will prefer dogs since more prefer dogs. (B) The next person he asks will prefer cats since more prefer dogs and it needs to be even. (C) The next person he asks will prefer cats since more prefer cats. (D) The preference to dogs or cats is not able to be determined based on the responses.” 

The materials do not provide opportunities for students to meet the full intent of the following standard:

• While students engage with 7.RP.1 (Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units.), students have no opportunities to work with unit rates as fractional lengths in regards to areas to meet the full intent of the grade-level standards.

The materials reviewed for Math Nation Grade 7 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

To determine the amount of time spent on major work, the number of units, the number of lessons, and the number of days were examined. Assessment days are included. Any lesson, assessment, or unit marked optional was excluded.

• The approximate number of units devoted to major work of the grade (including supporting work connected to the major work) is 4 out of 8, which is approximately 50%.

• The approximate number of lessons devoted to major work (including assessments and supporting work connected to the major work) is 81 out of 125, which is approximately 65%. 

• The approximate number of days devoted to major work of the grade (including supporting work connected to the major work) is 81 out of 125, which is approximately 65%.

A lesson-level analysis is most representative of the materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 65% of the materials focus on major work of the grade.

The materials reviewed for Math Nation Grade 7 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Materials are designed to connect supporting standards/clusters to the grade’s major standards/clusters. Examples of connections include:

• Unit 2, Lesson 8, 2.8.3 Exploration Activity, connects the supporting work of 7.G.6 (Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms) to the major work of 7.RP.2 (Recognize and represent proportional relationships between quantities). Students find and write equations for the total edge length, surface area, and volume while determining and explaining which relationships are proportional. “Here are some cubes with different side lengths. Complete each table. Be prepared to explain your reasoning. 1. How long is the total edge length of each cube? 2. What is the surface area of each cube? 3. What is the volume of each cube? 4. Which of these relationships is proportional? Explain how you know. 5. Write equations for the total edge length E, total surface area A, and volume V of a cube with side length s. Total edge length: E = ___ Total surface area: A = ___ Volume: V = ___.” Students are shown a picture of three cubes of side lengths 3, 5, and  9\frac{1}{2}. For Questions 1-3 students answer the question for the cubes listed above in addition to a cube of side length s.  

• Unit 3, Lesson 4, 3.4.3 Exploration Activity, connects the supporting work of 7.G.4 (Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle) to the major cluster of 7.NS.2 (Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers). Students use the formulas for area and circumference of a circle to answer questions about a running track. “The field inside a running track is made up of a rectangle that is 84.39 m long and 73 m wide, together with a half-circle at each end. 1. What is the distance around the inside of the track? Explain or show your reasoning. 2. The track is 9.76 m wide all the way around. What is the distance around the outside of the track? Explain or show your reasoning.” A diagram of the track is provided for students.

• Unit 8, Lesson 1, 8.1.5 Practice Problems, Question 3, connects the supporting work of 7.SP.6 (Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.) to the major work of 7.RP.1 (Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.). Students make predictions using probability to solve real-world problems. "A company tests two new products to make sure they last for more than a year. Product 1 had 950 out of 1,000 test items last for more than a year. Product 2 had 150 out of 200 last for more than a year. If you had to choose one of these two products to use for more than a year, which one is more likely to last? Explain your reasoning." 

• Unit 8, Lesson 16, 8.16.2 Exploration Activity, connects 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.NS.2 (Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers). Students use a random sample to solve proportion problems. “The track coach at a high school needs a student whose reaction time is less than 0.4 seconds to help out at track meets. All the twelfth graders in the school measured their reaction times. Your teacher will give you a bag of papers that list their results. 1. Work with your partner to select a random sample of 20 reaction times, and record them in the table. 2. What proportion of your sample is less than 0.4 seconds?  3. Estimate the proportion of all twelfth graders at this school who have a reaction time of less than 0.4 seconds. Explain your reasoning. 4. There are 120 twelfth graders at this school. Estimate how many of them have a reaction time of less than 0.4 seconds. 5. Suppose another group in your class comes up with a different estimate than yours for the previous question. a. What is another estimate that would be reasonable? b. What is an estimate you would consider unreasonable?”

The materials reviewed for Math Nation 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 major work to major work throughout the grade-level materials, when appropriate. Examples include:

• Unit 1, Lesson 6, 1.6.6 Practice Problems, Question 3, connects the supporting work of 7.G.A (Draw, construct, and describe geometrical figures and describe relationships between them) to the supporting work of 7.G.B (Solve real-life and mathematical problems involving angle measure, area, surface area, and volume). Students analyze the relationship between scaled figures, “Diego drew a scaled version of a Polygon P and labeled it Q. If the area of Polygon P is 72 square units, what scale factor did Diego use to go from P to Q? Explain your reasoning.” Students are given an image of Diego drawing.

• Unit 3, Lesson 6, 3.6.2 Exploration Activity, connects the supporting work of 7.G.A (Draw, construct, and describe geometrical figures and describe relationships between them) to the supporting work of 7.G.B (Solve real-life and mathematical problems involving angle measure, area, surface area, and volume). Students approximate the area of a home given a provided floor plan. “Here is a floor plan of a house. Approximate lengths of the walls are given. What is the approximate area of the home, including the balcony? Explain or show your reasoning.” Students are provided a floor plan of the house with numerous section measurements shown in feet.

• Unit 4, Lesson 3, 4.3.3 Exploration Activity, connects the major cluster of 7.RP.A (Analyze proportional relationships and use them to solve real-world and mathematical problems) to  the major work of 7.EE.B (Solve real-life and mathematical problems using numerical and algebraic expressions and equations). Students use proportional relationships to solve problems. “1. Tyler swims at a constant speed, 5 meters every 4 seconds. How long does it take him to swim 114 meters? 2. A factory produces 3 bottles of sparkling water for every 8 bottles of plain water. How many bottles of sparkling water does the company produce when it produces 600 bottles of plain water? 3. A certain shade of light blue paint is made by mixing 1\frac{1}{2} quarts of blue paint with 5 quarts of white paint. How much white paint would you need to mix with 4 quarts of blue paint? 4. For each of the previous three situations, write an equation to represent the proportional relationship.”

• Unit 6, Lesson 9, 6.9.5 Practice Problems, Question 3, connects the major cluster of 7.NS.A (Apply and extend previous understandings of operations with fractions) to the major work of 7.EE.A (Use properties of operations to generate equivalent statements). Students identify equations that match a tape diagram. “Select all the equations that match the diagram.” The equations listed are the following:  x+5 = 18; 183=x+5; C  3(x+5)=18;  x+5=\frac{1}{3}\cdot18;  3x+5=18.” The image of a tape diagram split into three sections with the equation, x + 5, is shown with the sum of all three pieces equal to 18.

The following connections are entirely absent from the materials:

• No connections are made between the supporting work of 7.G (Geometry) and the supporting work of 7.SP (Statistics & Probability). It is mathematically reasonable that the materials do not connect these two domains to each other. (Note: As shown above, there are instances where supporting clusters within the same domain have connections to each other.)

The materials reviewed for Math Nation 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.

Grade-level concepts related explicitly to prior knowledge from earlier grades along with content from future grades is identified and related to grade-level work in the Teacher Edition. Generally, explicit connections are found in the Course Guide or the Full Lesson Plan.  

Examples of connections to future grades include:

• Unit 2, Lesson 2, Full Lesson Plan, Lesson Narrative, connects 7.RP.2, 7.RP.2a, and 7.RP.2b to F-IF.1. “The purpose of this lesson is to introduce the concept of a proportional relationship by looking at tables of equivalent ratios. Students learn that all entries in one column of the table can be obtained by multiplying entries in the other column by the same number. This number is called the constant of proportionality…it prepares students for later work on functions, where they will think of x as the independent variable and y as the dependent variable.”

• Unit 2, Lesson 10, Full Lesson Plan, 2.10.3 Classroom activity, connects 7.RP.2 to 8.EE.6. “Students sort the graphs and justify their sorting schemes. Then, they compare the way they sorted their graphs with a different group. The purpose of this activity is to illustrate the idea that the graph of a proportional relationships is a line through the origin. Students will not have the tools for a formal explanation until grade 8.”

• Unit 7, Lesson 6, Full Lesson Plan, Lesson Narrative, connects 7.G.2 to 8.G.A and G-CO.B. “This lesson is the first in a series of lessons in which students create shapes with given conditions. During these lessons students think about what conditions are needed to determine a unique figure, in preparation for future work with congruence in grade 8 and high school.”

Examples of connections to prior knowledge include:

• Unit 2, Lesson 4, Full Lesson Plan, 2.4.3 Classroom activity, builds on 6.RP.3b and connects to 7.RP.2c. “As part of this activity, students calculate distance and speed. Students should know from grade 6 that speed is the quotient of distance traveled by amount of time elapsed, so they can divide 915 by 1.5 to get the speed. Students that do not begin the problem in that way can be directed back to the similar task in previous lessons to make connections and correct themselves. Once students have the speed, which is constant throughout this problem, they identify this as the constant of proportionality and use it to find the missing values.”

• Unit 4, Lesson 3, Full Lesson Plan, Lesson Narrative, builds on 6.RP.3 and connects to 7.RP.2. “In grade 6 students solved ratio problems by reasoning about scale factors or unit rates. In grade 7 they see the two quantities in a set of equivalent ratios as being in a proportional relationship and move towards using the constant of proportionality to find missing numbers…In this lesson students move toward solving problems involving proportional relationships by more efficient methods, especially by setting up and reasoning about a two-row table of equivalent ratios. This method encourages them to use the constant of proportionality rather than equivalent ratios.”

• Unit 5, Lesson 15, Full Lesson Plan, Lesson Narrative, builds on 6.EE.5 and connects to 7.NS.A and 7.EE.B. “The purpose of this lesson is to get students thinking about how to solve equations involving rational numbers. In grade 6, students solved equations of the form px=q and x+p=q and saw that additive and multiplicative inverses (opposites and reciprocals) were useful for solving them. However, that work in grade 6 did not include equations with negative values of p or q or with negative solutions. This lesson builds on the ideas of the last lesson and brings together the work on equations in grade 6 with the work on operations on rational numbers from earlier in grade 7.”