M1 Mid-Module Assessment Task - OpenCurriculum
NYS COMMON CORE MATHEMATICS CURRICULUM
Mid‐Module Assessment Task
M1
ALGEBRA II
Name Date 1. Geographers, while sitting at a café, discuss their field‐work site, which is a hill and a neighboring river bed. The hill is approximately 1050 feet high, 800 feet wide, with peak about 300 feet east of the western base of the hill. The river is about 400 feet wide. They know the river is shallow, no more than about twenty feet deep. They make the following crude sketch on a napkin, placing the profile of the hill and riverbed on a coordinate system with the horizontal axis representing ground‐level.
The geographers do not have with them at the café any computing tools, but they nonetheless decide to compute with pen and paper a cubic polynomial that approximates this profile of the hill and riverbed. a. Using only a pencil and paper, write a cubic polynomial function , that could represent the curve shown (here, represents the distance, in feet, along the horizontal axis from the western base of the hill, and is the height in feet of the land at that distance from the western base). Be sure that your formula satisfies 300 1050. Module 1: Date: © 2012 Common Core, Inc. All rights reserved. commoncore.org
Polynomial, Rational, and Radical Relationships 5/10/13
10
NYS COMMON CORE MATHEMATICS CURRICULUM b.
Mid‐Module Assessment Task
M1
ALGEBRA II
For the sake of ease, the geographers make the assumption that the deepest point of the river is halfway across the river (recall that the river is known to be shallow, with a depth of not more than 20 feet). Under this assumption, would a cubic polynomial provide a suitable model for this hill and riverbed? Explain.
2. Luke noticed that if you take any three consecutive integers, multiply them together, and add the middle number to the result, the answer always seems to be the middle number cubed. For example: 3 4 5 4 64 4 4 5 6 5 125 5 9 10 11 10 1000 10 a. In order prove his observation true, Luke writes down 1 2 3 2 . What answer is he hoping to show this expression equals? b. Lulu, upon hearing of Luke’s observation, writes down her own version with as the middle number. What does her formula look like? Module 1: Date: © 2012 Common Core, Inc. All rights reserved. commoncore.org
Polynomial, Rational, and Radical Relationships 5/10/13
11
NYS COMMON CORE MATHEMATICS CURRICULUM c.
Mid‐Module Assessment Task
M1
ALGEBRA II
Use Lulu’s expression to prove that adding the middle number to the product of any three consecutive numbers is sure to equal that middle number cubed.
3. A cookie company packages its cookies in rectangular prism boxes designed with square bases which have both a length and width of 4 inches less than the height of the box. a.
b.
Write a polynomial that represents the volume of a box with height inches. Find the dimensions of the box if its volume is equal to 128 cubic inches.
Module 1: Date: © 2012 Common Core, Inc. All rights reserved. commoncore.org
Polynomial, Rational, and Radical Relationships 5/10/13
12
NYS COMMON CORE MATHEMATICS CURRICULUM c.
Mid‐Module Assessment Task
M1
ALGEBRA II
After solving this problem, Juan was very clever and invented the following strange question: A building, in the shape of a rectangular prism with a square base, has on its top a radio tower. The building is 25 times as tall as the tower, and the side‐length of the base of the building is 100 feet less than the height of the building. If the building has a volume of 2 million cubic feet, how tall is the tower? Solve Juan’s problem.
Module 1: Date: © 2012 Common Core, Inc. All rights reserved. commoncore.org
Polynomial, Rational, and Radical Relationships 5/10/13
13
NYS COMMON CORE MATHEMATICS CURRICULUM A Progression Toward Mastery Assessment Task Item
1
2
Mid‐Module Assessment Task
M1
ALGEBRA II
STEP 1 Missing or incorrect answer and little evidence of reasoning or application of mathematics to solve the problem.
STEP 2 Missing or incorrect answer but evidence of some reasoning or application of mathematics to solve the problem.
STEP 3 A correct answer with some evidence of reasoning or application of mathematics to solve the problem, or an incorrect answer with substantial evidence of solid reasoning or application of mathematics to solve the problem.
STEP 4 A correct answer supported by substantial evidence of solid reasoning or application of mathematics to solve the problem.
a N‐Q.2 A‐APR.2 A‐APR.3 F‐IF.7c
Identifies zeros on graph. Uses zeros to write a factored cubic polynomial for H(x) without a leading coefficient.
Uses given condition 300 1050 to find a‐value (leading coefficient).
Writes a complete cubic model for H(x) in factored form with correct a‐value (leading coefficient).
b N‐Q.2 A‐APR.2 A‐APR.3 F‐IF.7c
Finds the mid‐point of the river.
Evaluates H(x) using the midpoint – exact answer is not needed, only approximation.
Determines if a cubic model is suitable for this hill and riverbed.
Justifies answer using H(midpoint) in explanation.
a A‐SSE.2 A‐APR.4
Answer does not indicate any expression involving n raised to an exponent of 3.
Answer involves a base involving n being raised to an exponent of 3, but does not choose a base of 2 .
Answers, 2 without including parentheses to indicate all of 2 is being cubed OR has another error that shows general understanding, but is technically incorrect.
Answers correctly as 2 .
b – c A‐SSE.2 A‐APR.4
Both parts b and c are missing OR incorrect OR incomplete.
Answer to part b is incorrect but student uses correct algebra as they attempt to show equivalence to OR the answer to part b is correct, but the student made major errors or
Part b is answered correctly as: 1 1 , but student made minor errors in showing equivalence to .
Answer is correctly written as: 1 1 AND the student correctly multiplied the left side and then combined like terms to show
Module 1: Date: © 2012 Common Core, Inc. All rights reserved. commoncore.org
Polynomial, Rational, and Radical Relationships 5/10/13
14
NYS COMMON CORE MATHEMATICS CURRICULUM
Mid‐Module Assessment Task
ALGEBRA II
was unable to show its equivalence to .
3
Determines an a – d expression for V(x). N‐Q.2 A‐SSE.2 A‐APR.2 A‐APR.3 A‐REI.1 A‐REI.4b
c N‐Q.2 A‐SSE.2 A‐APR.2 A‐APR.3 A‐REI.1 A‐REI.4b
Determines an expression for V(h) and sets it equal to the given volume.
M1
equivalence to
.
Sets V(x) equal to given volume.
Solves the equation understanding that only real values are possible solutions for the dimensions of a box.
States the 3‐dimensions of the box with proper units.
Simplifies the equation to reveal that is it exactly the same as the previous equation.
Solves the equation or states that the answer is exactly the same answer as in the previous example AND understands that only real solutions are possible for the height of a tower.
States the height of the tower with the proper units.
Module 1: Date: © 2012 Common Core, Inc. All rights reserved. commoncore.org
Polynomial, Rational, and Radical Relationships 5/10/13
15
NYS COMMON CORE MATHEMATICS CURRICULUM
Mid‐Module Assessment Task
M1
ALGEBRA II
Module 1: Date: © 2012 Common Core, Inc. All rights reserved. commoncore.org
Polynomial, Rational, and Radical Relationships 5/10/13
16
NYS COMMON CORE MATHEMATICS CURRICULUM
Mid‐Module Assessment Task
M1
ALGEBRA II
Module 1: Date: © 2012 Common Core, Inc. All rights reserved. commoncore.org
Polynomial, Rational, and Radical Relationships 5/10/13
17
NYS COMMON CORE MATHEMATICS CURRICULUM
Mid‐Module Assessment Task
M1
ALGEBRA II
Module 1: Date: © 2012 Common Core, Inc. All rights reserved. commoncore.org
Polynomial, Rational, and Radical Relationships 5/10/13
18
NYS COMMON CORE MATHEMATICS CURRICULUM
Mid‐Module Assessment Task
M1
ALGEBRA II
Module 1: Date: © 2012 Common Core, Inc. All rights reserved. commoncore.org
Polynomial, Rational, and Radical Relationships 5/10/13
19
