## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to identify linear and quadratic relationships in a realistic context: the number of tiles of different types that are needed for a range of square tabletops. In particular, this unit aims to identify and help students who have difficulties with: choosing an appropriate, systematic way to collect and organize data; examining the data and looking for patterns; finding invariance and covariance in the numbers of different types of tile; generalizing using numerical, geometrical or algebraic structure; and describing and explaining findings clearly and effectively.

- Level:
- Lower Primary, Upper Primary, Middle School, High School
- Grades:
- Kindergarten, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12
- Material Type:
- Assessment, Lesson Plan
- Provider:
- Shell Center for Mathematical Education, U.C. Berkeley
- Provider Set:
- Mathematics Assessment Project (MAP)
- Date Added:
- 04/26/2013

- License:
- Creative Commons Attribution Non-Commercial No Derivatives
- Media Format:
- Downloadable docs, Text/HTML

## Standards

Cluster: Mathematical practices

Standard: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x –1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x^2 + x + 1), and (x – 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build new functions from existing functions

Standard: (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build new functions from existing functions

Standard: Find inverse functions.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build new functions from existing functions

Standard: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build a function that models a relationship between two quantities

Standard: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build a function that models a relationship between two quantities

Standard: (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build a function that models a relationship between two quantities

Standard: Determine an explicit expression, a recursive process, or steps for calculation from a context.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build new functions from existing functions

Standard: Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2(x^3) or f(x) = (x+1)/(x-1) for x ≠ 1 (x not equal to 1).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build a function that models a relationship between two quantities

Standard: Write a function that describes a relationship between two quantities.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build a function that models a relationship between two quantities

Standard: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build new functions from existing functions

Standard: (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build new functions from existing functions

Standard: (+) Produce an invertible function from a non-invertible function by restricting the domain.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Building FunctionsCluster: Build new functions from existing functions

Standard: (+) Verify by composition that one function is the inverse of another.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Write a function that describes a relationship between two quantities.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2(x^3) or f(x) = (x+1)/(x-1) for x ‰äĘ 1 (x not equal to 1).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Determine an explicit expression, a recursive process, or steps for calculation from a context.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x -1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x^2 + x + 1), and (x - 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Produce an invertible function from a non-invertible function by restricting the domain.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: Find inverse functions.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Verify by composition that one function is the inverse of another.

Degree of Alignment: Not Rated (0 users)

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# Tags (11)

- Mathematics
- Algebra and Calculus
- Geometry and measures
- Algebra
- CCSS
- Common Core Math
- Common Core PD
- Geometry
- Measurement and Data
- ODE Learning
- Patterns and Sequencing

## Comments