7th Grade - Gateway 2
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Rigor & Mathematical Practices
Rigor & the Mathematical PracticesGateway 2 - Does Not Meet Expectations | 44% |
|---|---|
Criterion 2.1: Rigor and Balance | 5 / 8 |
Criterion 2.2: Math Practices | 3 / 10 |
The materials reviewed for Math Mammoth Grade 7, Light Blue Series, do not meet expectations for rigor and balance and practice-content connections. For rigor and balance, the materials do help students develop procedural skills and fluency, but they partially develop conceptual understanding. The materials partially provide opportunities for students to engage with multiple applications and partially balance the three aspects of rigor within the grade. The materials do not make meaningful connections between the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).
Criterion 2.1: Rigor and Balance
Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications.
The materials reviewed for Math Mammoth Grade 7, Light Blue Series, partially meet expectations for rigor. The materials give attention throughout the year to procedural skills and fluency. The materials partially develop conceptual understanding of key mathematical concepts, partially meet expectations for spending sufficient time working with engaging applications of mathematics, and partially balance the three aspects of rigor within the grade.
Indicator 2a
Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
The materials reviewed for Math Mammoth Grade 7, Light Blue Series, partially meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. The materials do not provide opportunities to develop conceptual understanding throughout the grade level as students are provided the procedure to solve problems during the introduction of conceptual understanding.
The materials do not provide opportunities for students to develop conceptual understanding throughout the grade level. Examples include, but are not limited to:
Worktext 7-A, Chapter 1: The Language of Algebra, Properties of the Four Operations, Question 1, students compare expressions to find which ones are equivalent. “Are the two expressions in each box equivalent? That is so they have the same value for any value of x? Give c some test values to check.”The following expressions are given: a. c+5, 5+c b. c-5, 5-c c. c÷6, 6÷c d. 5c, c⋅5 . This question does not provide an opportunity for students to build a conceptual understanding of 7.EE.1(Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.) as students are guided to check different solutions to determine if expressions are equivalent.
Worktext 7-A, Chapter 2: Integers, Addition and Subtraction on the Number Line, a box is provided with the following information. “What about subtracting a negative number? Here is a way to think about 3 − (−2) on the number line. Imagine you are standing at 3. Because of the subtraction sign, you turn to the left and get ready to take two steps. However, because of the additional minus sign in front of the 2, you have to take those steps BACKWARDS—to the right! So, because you ended up taking those 2 steps to the right, in effect you have just performed 3 + 2! Quite surprising, but true: that double minus sign in 3 − (−2) turns into a plus!” Question 16, “Draw a number line jump for each subtraction, using the method just explained. a. 1-(-2) = ___” This question does not provide an opportunity for students to build a conceptual understanding of 7.NS.1 (Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.)as students are directed to use the method previously mentioned to get the correct answer.
Worktext 7-B, Chapter 6: Ratios and Proportions, Why Cross-multiplying Works, Instructional box, “Recall that if we multiply both sides of an equation by the same number, the two sides are equal. In a proportion, we have two different numbers in the denominators, we can first multiply both sides of the proportion by the one denominator and then by the other, or we can cross-multiply. Cross-multiplying is in reality just a shortcut for doing those two separate multiplications at the same time. Let’s solve the proportion below by multiplying both sides first by the one denominator, then by the other. Cross-multiplying is not a ‘magic trick’, but simply a shortcut based on mathematical principles.” Students are then shown a demonstration of how to solve a proportion without cross-multiplying. This instruction does not provide an opportunity for students to build a conceptual understanding of 7.NS.2 (Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers) as students are explained why cross-multiplying works.
Indicator 2b
Materials give attention throughout the year to individual standards that set an expectation for procedural skill and fluency.
The materials reviewed for Math Mammoth Grade 7, Light Blue Series, meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
Materials develop procedural skills and fluency throughout the grade level. Examples include:
Worktext 7-A, Chapter 4: Rational Numbers, Rational Numbers, Question 16, students convert fractions to decimals using long division. “Write as decimals, using a line over the repeating part (if any). Use long division. d. 2$$\frac{7}{16}$$“ . This activity provides students with an opportunity to develop procedural skills and fluency with 7.NS.2d (Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats).
Worktext 7-B, Chapter 7: Percent, Percent Equations, Question 2, students select the correct equation for a world problem and then solve it. “A computer is discounted by 25\%, and now it costs $$\$576$$. Let p be its price before the discount. Select the equation that matches the statement above and solve it.” The equations given are the following: p+0.25p=576, p-0.25p=576, 0.25p=576. This activity provides students with an opportunity to develop procedural skills and fluency with 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 specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach).
Worktext 7-B, Chapter 8: Geometry, Chapter 8 Mixed Review, Question 8, students simplify several expressions. “Simplify the expressions. a. 2+w+11+w+2w” This activity provides students with an opportunity to develop procedural skills and fluency with 7.EE.1 (Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients).
The materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:
Worktext 7-A, Chapter 1: The Language of Algebra, Simplifying Expressions, Question 8, students simplify several expressions. “Simplify the expressions. a. 5p+8p-p". This activity provides students with an opportunity to independently demonstrate the procedural skill and fluency of 7.EE.1 (Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients).
Worktext 7-A, Chapter 5: Equations and Inequalities, Two-Step Equations: Practice, Question 4, students solve two-step equations. “Solve. Check your solutions (as always!) a. 20-3y=65". This activity provides students with an opportunity to independently demonstrate the procedural skill and fluency of 7.EE.B (Solve real-life and mathematical problems using numerical and algebraic expressions and equations).
Worktext 7-B, Chapter 6: Ratios and Proportions, Chapter 6 Mixed Review, Question 4, students solve mathematical problems using the four operations. “Find the value of the expressions using the correct order of operations. a. 5⋅\frac{2}{-10} b. -\frac{12}{-4}+7” This activity provides students with an opportunity to independently demonstrate the procedural skill and fluency of 7.NS.3 (Solve real-world and mathematical problems involving the four operations with rational numbers).
Indicator 2c
Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.
The materials reviewed for Math Mammoth Grade 7, Light Blue Series, partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. The materials include multiple opportunities for students to engage with and independently demonstrate routine applications of the mathematics throughout the grade level. However, the materials include few opportunities for students to engage with and independently demonstrate non-routine applications of the mathematics throughout the grade level.
Examples of students engaging in routine application of grade-level skills and knowledge, within instruction and independently, include:
Worktext 7-A, Chapter 1: The Language of Algebra, The Distributive Property, Question 16, students solve a multi-step real-life problem with rational numbers and construct a simple equation by reasoning about quantities. “The Larson family are planning their new house. It is going to be 25 ft on one side and have a garage that is 15 ft wide, but they have not decided on the length of the house yet. a. If the total area of the house + garage is limited to 1200 square feet, how long can the house be? b. Write a single equation for the question above. Write it in the form ‘(formula of area) = 1200.’ You do not have to solve the equation–just write it.” Students are provided a diagram of a rectangle split in two vertically, one section is labeled garage the other is labeled house. One side of the rectangle is labeled 25ft and the other side is labeled 15ft and x. This problem allows students to apply the mathematics of 7.EE.B (Solve real-life and mathematical problems using numerical and algebraic expressions and equations.) in a routine application problem independently.
Worktext 7-A, Chapter 3: Solving One-Step Equations, Multiplication and Division Equations, Question 6, students write an equation for a word problem and then solve it. “Write an equation for each situation. Then solve it. Do not write the answer only, as the main purpose of this exercise is to practice writing equations. a. A submarine was located at a depth of 500 ft. There was a shark swimming at 1/6 of that depth. At what depth is the shark?” This problem allows students to apply the mathematics of 7.EE.4 (Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities) in a routine application problem independently.
Worktext 7-B, Chapter 8: Geometry, Chapter 8 Mixed Review, Question 11, students find the area of a rectangle after it is enlarged. “A rectangle with sides of 2 1/4 in. and 3 in. is enlarged by a scale factor of 3.5. Find the area of the resulting rectangle to the tenth of a square inch.” This problem allows students to apply the mathematics 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.) in a routine application problem independently.
The materials provide few opportunities for students to engage with and independently demonstrate non-routine applications throughout the grade level. An example where a student would engage in a non-routine application is shown below.
Worktext 7-A, Chapter 2: Integers, Adding or Subtracting Several Integers, Question 8, students create a story based on an expression. “Write a story about money to match the expression -2 - (-10) + (-7) - 4. “ This problem allows students to apply the mathematics of 7.NS.3 (Solve real-world and mathematical problems involving the four operations with rational numbers) in a non-routine application problem.
Indicator 2d
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.
The materials reviewed for Math Mammoth Grade 2, Light Blue Series, partially meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. The materials do not balance all three aspects of rigor, as there is an over-emphasis on procedural skills and fluency.
The materials provide some opportunities for students to develop conceptual understanding, procedural skills and fluency, and application separately throughout the grade and some opportunities where multiple aspects of rigor are used to support student learning and mastery of the standards. However, there are multiple lessons where one aspect of rigor is emphasized.
Examples of conceptual understanding, procedural skill and fluency, and application presented separately in the materials include:
Worktext 7-A, Chapter 2: Integers, Subtraction of Integers, Question 1, students develop conceptual understanding of adding and subtracting integers by using counters. “Model Subtraction with counters. You may need to add positive-negative pairs before subtracting. a. -4 - (-2) = _____” Students are shown four negative counters. This activity provides students with an opportunity to develop a conceptual understanding with 7.NS.1 (Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers).
Worktext 7-B, Chapter 6: Ratio and Proportions, Unit Rates, Question 3, students find the unit rate based on given situations. “Write the unit rate as a complex fraction, and then simplify it. a. Lisa can make three skirts out of 5 ½ yards of material. Find the unit rate for one skirt.” This problem allows students to apply the mathematics 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.) in a routine application problem independently.
Worktext 7-B, Chapter 7: Percent, Solving Basic Percentage Problems, Question 7, students solve a series of percent tax problems. “Find the final price when the base price and sales tax rate are given. This is a mental math workout, so do not use a calculator! a. Bicycle: $$\$100$$; 7$$\%$$ sales tax. Tax to add: \$____ Price after tax: \$____” This activity provides students with an opportunity to develop procedural skills and fluency with 7.RP.3 (Use proportional relationships to solve multistep ratio and percent problems).
Examples of students having opportunities to engage in problems that use two or more aspects of rigor include:
Worktext 7-A, Chapter 1: The Language of Algebra, Properties of the Four Operations, Question 9, students write and simplify an expression based on an illustration. “Write an expression from the illustration and simplify it.” The illustration shows a line divided into five equal parts with an x under each part. Students develop conceptual understanding and build procedural skills and fluency for 7.EE.1 (Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients) as they form expressions based on the illustration and simply them.
Worktext 7-A, Chapter 5: Equations and Inequalities, Word Problems, Question 2, students solve for the width of a rectangle knowing only the length and the perimeter. “Write an equation for the following problem and solve it. The perimeter of a rectangle is 144 cm. Its length is 28 cm. What is its width?” Students develop conceptual understanding and application of 7.EE.4 (Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.) as they solve a word problem for an unknown based on their understanding of perimeter.
Criterion 2.2: Math Practices
Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).
The materials reviewed for Math Mammoth Grade 7, Light Blue Series, do not meet expectations for practice-content connections. The materials support the intentional development of MP6 (attend to precision) and attend to the specialized language of mathematics. The materials partially support the intentional development of MP3 (Construct viable arguments and critique the reasoning of others). The materials do not support the intentional development of MPs 1, 2, 4, 5, 7, and 8. Additionally, the materials do not explicitly identify the mathematical practices in the context of individual lessons, so one point is deducted from the score in indicator 2e to reflect the lack of identification.
Indicator 2e
Materials support 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 reviewed for Math Mammoth Grade 7, Light Blue Series, do not 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. There is no intentional development of MP1 to meet its full intent in connection to grade-level content. The Standards of Mathematical Practice are not explicitly identified in the context of the individual lessons for teachers or students. As a result of this one point is deducted from the scoring of this indicator.
MP1 is not intentionally developed to meet its full intent as students have limited opportunities to make sense of problems and persevere in solving them. Examples include, but are not limited to:
Worktext 7-A, Chapter 1: The Language of Algebra, The Order of Operations, Question 8, “Rewrite each expression using a fraction line, then simplify. Compare the expression in the top row with the one below it. Hint: Only what comes right after the “÷” sign goes into the denominator. a. 2÷5⋅4 “ This problem does not provide students with an opportunity to make sense of problems and persevere in solving them, since hints are provided to help students.
Worktext 7-A, Chapter 2: Integers, Distance and More Practice, Question 6, students determine if the order of numbers is important when considering distance. “What happens if you calculate the distance between two numbers m and n as |n-m| instead of |m-n|? In other words, what happens if you reverse the order of the numbers in the subtraction? Investigate this by checking several pairs of numbers. For example, what happens if you calculate the distance between 18 and 11 as |11-18| instead of |18-11|? Try some pairs of negative numbers, as well. Then try some pairs where one number is positive and the other is negative. Does the formula still work?” This problem does not provide students with an opportunity to make sense of problems and persevere, since it explains to students strategies to investigate.
Worktext 7-B, Chapter 6: Ratios and Proportions, Proportional Relationships, Question 3, students are asked to consider how proportional relationships are represented in tables, graphs and equations. “Now consider the plots, the equations, and the tables of values of the six items in the previous exercise. How do the equations and plots of the variables that are in proportion differ from those that aren’t? If you cannot tell, check the next page.” This problem does not provide students with an opportunity to make sense of problems and persevere, since it tells students they can check the next page for the answer.
MP2 is connected to grade-level content, and intentionally developed to meet its full intent. Students reason abstractly and quantitatively as they work independently throughout the units. Examples include:
Worktext 7-A, Chapter 2: Integers, Dividing Integers, Question 4, students must write an equation developed from an understanding of positive and negative numbers. “In a math game, you get a negative point for every wrong answer and a positive point for every correct answer. Additionally, if you answer in 1 second, your negative points from the past get slashed in half! Angie had accumulated 14 negative points and 25 positive points in the game. Then she answered a question correctly in 1 second. Write an equation for her current ‘point balance.’”This question attends to the full intent of MP2, reasoning abstractly and quantitatively as students write an equation about the current point balance of a game.
Worktext 7-B, Chapter 6: Ratios and Proportions, Chapter 6 Review, Question 10, students reason quantitatively to determine if quantities are proportional. “Using a pre-paid internet service you get a certain amount of bandwidth to use for the amount you pay. The table shows the prices for certain amounts of bandwidth. a. Are these two quantities in proportion? Explain how you can tell that. b. If so, write an equation relating the two and state the constant of proportionality.” Students are provided a table of the bandwidth ranging from 1G to 25G and the price ranging from $$\$10$$ to $$\$50$$. This question attends to the full intent of MP2, reasoning abstractly and quantitatively as students use the information from the table to explain if the two quantities are in proportion and to write an equation that relates the two quantities.
Worktext 7-B, Chapter 7: Percent, Percentage of Change: Applications, Question 3, students answer questions about the side length and area of a square after the sides increase by a scale factor. “The sides of a square are increased by a scale factor of 1.15. a. By what percentage does the length of each side increase? b. What is the percentage of increase in area? Hint: Make up a square using an easy number for the length of side. c. (Challenge) Would your answers to (a) and (b) change if the shape were a rectangle? A triangle?” This question attends to the full intent of MP2, reasoning abstractly and quantitatively as students reason about how an increase in side length, increases the area percentage.
Indicator 2f
Materials support 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 reviewed for Math Mammoth Grade 7, Light Blue Series, partially 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. Although support for the intentional development of constructing viable arguments are found throughout the materials, support for the intentional development of critiquing the reasoning of others is limited to certain chapters.
Materials provide support for the intentional development of MP3 by providing opportunities for students to construct viable arguments in connection to grade-level content. Examples include:
Worktext 7-A, Chapter 4: Rational Numbers, Adding and Subtracting Rational Numbers, Question 11, students construct viable arguments as they explain a real-life situation based on an expression. “Explain a real-life situation for the sum $$ -\$50+(-\$12.90)$$.” This question attends to MP3, constructing viable arguments as students explain a real-life situation based on the sum.
Worktext 7-A, Chapter 5: Equations and Inequalities, Word Problems and Inequalities, Question 2, students explain their solution set to a word problem using their own words. “Jeannie earns $$\$350$$ per week plus $$\$18$$ for each hour of overtime that she works. How many hours of overtime does she need to work if she wants to earn at least $$\$500$$? Write an inequality and solve it. Plot the solution set on a number line. Lastly, explain the solution set in words.” This question attends to MP3, constructing viable arguments as students explain the solution set to an inequality that they wrote and solved.
Worktext 7-B, Chapter 8: Geometry, Area and Perimeter Problems, Question 5, students justify their answer for whether or not a unique trapezoid is formed from given conditions. “The two parallel sides of a trapezoid measure 12 cm and 9 cm, and its altitude is 7 cm. a. Draw a trapezoid using this information, either on paper or in drawing software. b. Do the specified dimensions determine a single, unique trapezoid, or is it possible to draw more than one shape of trapezoid that satisfies those dimensions? Justify your answer.” This question attends to MP3, constructing viable arguments as students justify their answer about what kind of trapezoid could be created based on the dimensions given.
Materials provide some support for the intentional development of MP3 by providing limited opportunities for students to critique the reasoning of others in connection to grade-level content.
Examples include, but are not limited to:
Worktext 7-B, Chapter 6: Ratios and Proportions, Solving Proportions: Cross Multiplying, Question 7, students perform error analysis as they explain what is wrong with a proportion. “Jane wrote a proportion to solve the following problem. Explain what is wrong with her proportion and correct it. Then solve the corrected proportion. Also, check that your answer is reasonable. Twenty kilograms of premium dog food cost $51. How much would 17kg cost?” An image labeled Jane’s proportion is provided for students. This question attends to MP3, critiquing the reasoning of others as students correct another person's error.
Worktext 7-B, Chapter 11: Statistics, Comparing Two Populations, Question 7, students critique the reasoning of others as they explain if an organization's way of selecting a sample is good. “An organization that helps teenagers with drug problems has set up a telephone hot line for teens to call in to discuss their problems. After a few months of operations, the organization wants to evaluate the effectiveness of their service. Since they don’t usually get as many calls on Tuesdays, they decide to choose a particular Tuesday to ask each teen at the end of the call to answer a few questions about how the service has helped. Is this a good method for selecting a sample? Explain.” This question attends to MP3, critiquing the reasoning of others as students explain if an organization's way of selecting sample is good or not.
Worktext 7-B, Chapter 11: Statistics, Chapter 11 Mixed Review, Question 13, students critique the reasoning of others as they explain why a sampling method is biased. “Sam is studying how well the people in his city like the paintings of the Romantic era. He is planning to stand on a certain street corner near his home and ask passersby if they would like to take part in his study. Explain why his sampling method is biased.” This question attends to MP3, critiquing the reasoning of others as students explain why another person's sampling method is biased.
Indicator 2g
Materials support the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for Math Mammoth Grade 7, Light Blue Series, do not 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. There is no intentional development of MP4 and MP5 to meet its full intent in connection to grade-level content.
MP4 is not intentionally developed to meet its full intent as students have limited opportunities to model with mathematics. Examples include, but are not limited to:
Worktext 7-A, Chapter 1: The Language of Algebra, Expressions and Equations, Question 5, students write an expression and pick an equation based on a real-world scenario. “a. Ann is 5 years older than Tess, and Tess is n years old. Write an expression for Ann’s age. b. Let A be Alice’s age and B be Betty’s age. Find the equation that matches the sentence ‘Alice is 8 years younger than Betty.’ Hint: give the variables some test values.” The equation choices are the following: A=8-B, A=B-8, and B=A-8. This problem does not provide students with an opportunity to model the situation with an appropriate representation as students are told to write an expression, with the variables given and to find the equation that matches the sentence.
Worktext 7-A, Chapter 2: Integers, Addition of Integers, Question 4, students fill in a sentence to compare how an expression is modeled with two different methods. “4. Compare how −8 + 6 is modeled on the number line and with counters. a. On the number line, −8 + 6 is like starting at _____, and moving _____steps to the _____________, ending at _____. b. With counters, −8 + 6 is like _____ negatives and _____ positives added together. We can form _____ negative-positive pairs that cancel each other out, and what is left is ____ negatives.” This problem does not provide students with an opportunity to model the situation with an appropriate representation as students are filling in blanks instead of modeling to explain how the number line compares to counters when used to solve addition problems with integers.
Worktext 7-A, Chapter 4: Rational Numbers, Multiply and Divide Rational Numbers 2, Question 10, students create a written problem for each multiplication expressions and then solve them. “Give a real-life context for each multiplication. Then solve. I have already done the first two for you. Hint: The area of a rectangle, the length resulting from stretching or shrinking a dimension, a fractional part, and a percentage of a quantity are all calculated by multiplying. c. (9/10)· 2,100m d. 0.65 · 19.90” This problem does not provide students with an opportunity to model the situation with an appropriate representation as students are given hints and examples that they could use.
MP5 is not intentionally developed to meet its full intent as students have limited opportunities to choose tools strategically. Examples include, but are not limited to:
Worktext 7-A, Chapter 1: The Language of Algebra, Simplifying Expressions, Puzzle Corner, students write an expression and make a table to figure out how many coins a person has. “a. What is the total value, in cents, if Ashley has n dimes and m quarters? Write an expression. b. The total value of Ashley’s coins is 495 cents. How many dimes and quarters can she have? Hint: make a table to organize the possibilities.” This problem does not provide students with an opportunity to choose tools strategically as the problem gives students the hint of making a table and does not allow them to consider the tools available.
Worktext 7-A, Chapter 4: Rational Numbers, Adding and Subtracting Rational Numbers, Question 6, students find the distance between two numbers. “Find the distance between the two numbers. The number lines above can help. a. -0.8 and -2.2 b. 0.9 and -1.3” This problem does not provide students with an opportunity to choose tools strategically as the problem provides students with the tool they can use to help (number line), and does not allow them to consider the tools available.
Worktext 7-A, Chapter 5: Equations and Inequalities, Using the Distributive Property, Question 2, students solve a two-step equation. “Solve in two ways: (i) by dividing first and (ii) by distributing the multiplication over the parentheses first. a. 6(x-7)=72” This problem does not provide students with an opportunity to choose the method of solving as the problem recommends the ways for students to solve it.
Indicator 2h
Materials attend to 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 reviewed for Math Mammoth Grade 7, Light Blue Series, 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.
Materials provide support for the intentional development of MP6 by providing opportunities for students to attend to precision in connection to grade-level content. Examples include:
Worktext 7-A, Chapter 3: Solving One-Step Equations, Constant Speed, Question 8, students find the average speed for several scenarios in the given units. “Find the average speed in the given units. a. A duck flies 3 miles in 6 minutes. Give your answer in miles per hour. b. A lion runs 900 meters in 1 minute. Give your answer in kilometrs per hour.” This problem intentionally develops MP6 as it requires students to accurately calculate the average speed in the units given.
Worktext 7-A, Chapter 4: Rational Numbers , Rational Numbers, Question 3, students convert fraction to decimals. “Form a fraction from the two given integers. Then convert it into a decimal. a. 8 and 5 b. -4 and 10 c. 89 and -100” This problem intentionally develops MP6 as it requires students to be precise in their calculation in order to convert the fraction to decimals correctly.
Worktext 7-B, Chapter 6: Ratios and Proportions, Floor Plans, Questions 1, 2, and 3, students use a scale drawing to answer questions about items' dimensions in reality. “1. This room is drawn at a scale of 1in : 4ft. Measure dimensions asked below from the picture and then calculate the actual (real) dimensions. a. the bed b. the desk 2. What is the area of this room in reality? 3. In the middle of the plan for the room, draw a table that in reality measure 3.5 ft x 2.5 ft.” This problem intentionally develops MP6 as it requires students to express the length of objects with a degree of precision.
Materials provide support for attending to the specialized language of mathematics. Examples include:
Worktext 7-A, Chapter 2: Integers, Integers, Question 7, students are tasked with writing an expression for a word or phrase using symbols. “Write using symbols, and simplify if possible. a. the opposite of 6 b. the opposite of -11 c. the opposite of the absolute value of 12 d. the absolute value of negative 12 e. the opposite of the sum 6 + 8 f. the opposite of the difference 9 - 7 g. the absolute value of the opposite of 8 h. the absolute value of the opposite of -2”. This problem attends to the specialized language of mathematics as students translate written phrases into written expressions and simplify (if possible) those expressions.
Worktext 7-A, Chapter 5: Equations and Inequalities, Two-Step Equations: Practice, Question 5, students solve a word problem involving a quadrilateral. “Solve each problem below in two ways: write an equation, and use logical reasoning/mental math. a. A quadrilateral has three congruent sides. The fourth side measures 1.4 m. If the perimeter of the quadrilateral is 7.1 meters, what is the length of each congruent side? This problem attends to the specialized language of mathematics as students use the definition of congruency to solve the problem.
Worktext 7-B, Chapter 6: Ratios and Proportions, Scaling Figures, Question 6, students develop an understanding of the scale factor and scale ratio in a specific situation. Notes within the lesson define scaling and scale ratio, it also says the following: “Here is a way to keep the two very similar-sounding terms straight: The scale ratio is a ratio of two numbers (like 3:1), but the scale factor is a single number (such as 3).” Question 6, “a. Find the scale factor from the smaller to the larger parallelogram. b. What is the scale ratio?” A picture of two parallelograms of different sizes is provided. The larger parallelogram has side lengths of 19 cm and 14 cm. The smaller parallelogram has 6 cm on the shorter side. This problem attends to the specialized language of mathematics as students use the definition of scale factor and ratio to provide a correct solution.
Indicator 2i
Materials support 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 the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for Math Mammoth Grade 7, Light Blue Series, do not 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 provide opportunities for students to engage with structure and repeated reasoning, but do not allow students to use the structure or the repeated reasoning to formulate their ideas.
MP7 and MP8 are not intentionally developed to meet their full intent in connection to grade-level content as students have limited opportunities to look for and make use of structure and look for and express regularity in repeated reasoning. Examples include, but are not limited to:
Worktext 7-A, Chapter 1: The Language of Algebra, The Distributive Property, Question 1, students use the structure of a line segment to recognize the repeated pattern involved in the distributive property. “Write an expression for the repeated pattern in the model. Then multiply the expression using the distributive property.” In part b, students are provided a line segment with four s’s under the blue portion of the line segment each s is followed by the number 11 which is under the red portion of the line segment. Above Question 1 in an instructional box, students are provided a model of how to write and multiply an expression using line segments. “Here is a way to model the distributive property using line segments. The model shows a pattern of line segments of length x and 1 repeated four times. In symbols, we write 4⋅(x+1). However, it is easy to see that the total length can also be written as 4x+4. Therefore, 4⋅(x+1)=4x+4. Students are not provided with the opportunity to make use of the structure of the line segment or the repeated reasoning it provides since the problem was modeled for them before they attempted it.
Worktext 7-A, Chapter 2: Integers, Multiplying Integers, Question 5, students use the structure of a series of equations to complete a pattern. “Complete the patterns. …b. -5⋅3=____ -5⋅2=____ -5⋅1=____ -5⋅0=____ -5⋅(-1)=____ -5⋅(-2)=____ -5⋅(-3)=____ -5⋅(-4)=____ In the pattern above, the product (answers) increase by ____ in each step! … The patterns in the products show that to be consistent, a negative times a negative must be a positive.” Students are not provided with the opportunity to share their thinking about what they notice while continuing the pattern through repeated reasoning and comparing the columns.
Worktext 7-B, Chapter 6: Ratios and Proportions, Proportional Relationships, Question 1, students fill in a table to determine whether two variables are in direct variation or not. “Fill in the table of values and determine whether the two variables are in direct variation. a. y=3x” Students are given a table with the x values ranging from -3 to 4 increasing by increments of 1, and the y values of the table are blank. Above Question 1 in an instructional box, students are provided a model of how to check to see if two variables are in direct variation. “You can check to see if two variables are in direct variation in several different ways. Here is one way. (1) Check to see if the values of the variables are in direct variation. If you double the value of one, does the value of the other double also? If one quantity increases by 5 times, does the other do the same?” This question is followed by an example of a relationship presented in a table that is not in direct variation accompanied by an explanation of why the values in the table do not work. Students are not provided with the opportunity to make use of the structure of the table to make their own generalizations about direct variation.