8th Grade - Gateway 2

The instructional materials do not meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics without losing focus on the major work of each grade. Overall, there is little evidence of the opportunity to work with engaging applications of the mathematics. There are few non-routine problems throughout the year. Word problems are present in the materials, but the context has limited bearing on the mathematics. There are ten Problem Solving lessons designed to "isolate and and focus on (problem solving) strategies."

Cluster 8.F.B involves students using functions to model relationships between quantities.

• Teacher Resources Part 2 Unit 1 Lessons 15-19, 21, and 22 are identified as using functions to model relationships between quantities. Lessons 15 through 19 in the Teacher Resources explicitly instruct students on how to find the y-intercept from a graph, equation, table, or ordered pairs and then write the equation of a line in slope-intercept form. Lesson 21 systematically instructs students on how to solve five different word problems, two of them extension problems, using linear functions in the first quadrant by asking numerous questions: "How much do you have to pay to rent the e-bike for 1 hour? For 5 hours?;" "What is the y-intercept of the line?;" and "How are the slope and the cost related?" Lesson 22 gives three opportunities for students to qualitatively describe linear pieces of a graph, but these problems are based on a teacher-led example that shows students a process for completing the problems.

Standard 8.EE.8c has students solving real-world and mathematical problems leading to two linear equations in two variables.

• In Teacher Resources Part 2 Unit 5, Lessons 52 and 55 are identified as solving real-world and mathematical problems leading to two linear equations in two variables. Lesson 52 directly instructs students on solving a word problem by graphing the two equations and finding an intersection point in the first quadrant. Two extension word problems are also given that lead students to graphing and finding a solution at the intersection point.
• Pages 143-144 in the Assessment and Practice book for Part 2 has students writing two formulas for word problems without solving, solving two word problems by graphing the equations created by the students, solving four word problems by graphing equations that may contain fractions or decimals, and explaining why it might be difficult to find the exact intersection when fractions and decimals are included. In these problems, students are following a process established by the problems and exercises in the Teacher Resources book.
• In Lesson 55 of Unit 5 of Teacher Resources Part 2, the teacher is instructed to write a four-step process on the board that students can use in order to solve word problems that result in two linear equations in two variables. The contexts presented do not have bearing on solving the problems, and the students are shown examples of how to "translate" the word problems into equation through the use of certain key words.

Problem Solving lesson PS8-7, found on G8-44, focuses on using logical reasoning to understand and reproduce a proof of the Pythagorean Theorem. Problems are presented with extensive scaffolding followed by exercises that are similar to those in the lesson. There are two BLMs: Proving the Pythagorean Theorem 1 and Proving the Pythagorean Theorem 2. Neither sheet engages students with an application of the mathematics or an opportunity to explain the proof of the Pythagorean Theorem.

The instructional materials reviewed for Grade 8 do not meet the expectations for carefully attending to the full meaning of each practice standard. The publisher rarely addresses the Mathematical Practice Standards in a meaningful way.

The materials identify examples of the Standards for Mathematical Practice (MPs), so the teacher does not always know when a MP is being carefully attended to. MPs are marked throughout the curriculum, but sometimes the problems are routine problems that do not cover the depth of the Math Practices. Many times the MPs are marked where teachers are doing the work.

Examples where the material does not meet the expectation for the full meaning of the identified MP:

• MP1: In Teacher Resources Part 1 Unit 4 Lesson 32, the directions ask the teacher to “Have a volunteer circle all the x’s. How many x’s are being added in total?”, and in Teacher Resources Part 1 Unit 4, Lesson 33 specifically states to, “Encourage students to check their answers for one or two questions by replacing the variable x with a value such as x = 1 in both the given expression and its simplified expression. Students should find that they get the same result for both expressions. If not, they should look for a mistake in their work.” In these two examples, students do not have to make sense of problems to obtain the expressions they are trying to simplify because the expressions are given to them, and the level of perseverance required of the students is minimal due to the complexity of the problems being solved.
• MP1: While the materials attach MP1 to many lessons, sometimes the extent of scaffolding takes away the student's opportunity to reason and persevere. For example, in Teacher Resources Part 2 Unit 1 Lesson 15, MP1 is claimed four times. Students are given several reminders by the teacher when finding the y-intercept. The teacher says, “Remind students that because the slope is equal between every two points on a straight line, we can say the slope of this line is ½.” Later in the lesson the teacher again, “remind(s) students that, in an increasing graph, as the x-coordinates get bigger, the y-coordinates get bigger too” and “remind students that they can draw a line using two points.” Due to the overabundance of teacher reminders, students do not have to make sense or persevere while solving these problems.
• MP2: In Lesson 50 of Unit 5 in Teacher Resources Part 2, there are multiple instances where this MP is identified, but none of the problems contain a context that would enable students to engage in quantitative reasoning. Students do not have to attend to the meaning of quantities or create any representations for the problems presented because the abstract equations are given in the problem, and there are no contexts in which the students have to interpret, or contextualize, the abstract solutions.
• MP4: In Lesson 27 of Unit 4 in Teacher Resources Part 1, there are two references to this MP. One set of exercises has students "write an expression for the cost of renting skates," and the other reference has teachers "work through one example with students, modeling each step on the board." In these two examples, students are not creating a mathematical model, and due to the teacher completing an example for the students, students are not getting to make assumptions or approximations in a complex situation or identify important quantities and represent their relationships.
• MP4: In Teacher Resources Part 2 Unit 1, Lesson 19 cites MP4 when it has students substituting given values of m and b in the equation y=mx+b. In this problem, a mathematical model is given to the students, and as with many other problems where MP4 is noted, students do not have the opportunity to make assumptions or approximations in a complex situation, identify important quantities and represent their relationships, draw conclusions, or interpret the results of a problem and make improvements if needed.
• MP5: In Teacher Resources Part 1 Unit 3 Lesson 14, students are working with corresponding angles and parallel lines. Activity 1 is labeled with MP5; however, the directions specifically instruct students to use The Geometer’s Sketchpad. This direct instruction on what tool students should use happens with activities in other lessons in this Unit. To reach the full meaning of this MP, students should be familiar with various tools appropriate for their grade and be able to make sound decisions about when each of the tools might be helpful, recognizing both the insight to be gained and their limitations.
• MP6: In Teacher Resources Part 2 Unit 2, Lesson 20 has students working with parallel lines and transversals. The materials state, “Extend the slanted sides of all three triangles into lines and label them as shown below. ASK: Are any of these lines parallel? How can you tell? To prompt students to see the answer, highlight the horizontal line as shown below and label it line t.” Though this does require the teacher to be as precise as possible, it does not have the student working with the lines, so it does not attain the full meaning of having students attend to precision.