## 3rd Grade

- Operations & Algebraic Thinking
- Number & Operations in Base Ten
- Number & Operations - Fractions
- Measurement & Data
- Geometry

**Represent and solve problems involving multiplication and division.**CCSS.MATH.CONTENT.3.OA.A.1

To craft tools, students will need to determine how many groups of sticks and planks they need by interpreting products of whole numbers. For example, they will say, "To make three swords, I need 3 groups of 2 planks and 3 groups of 1 stick." |

**Multiplication Towers**

This is the two’s times tables. |

**Commutative Property of Multiplication**

This is 3x5=15 and 5x3=15. |

**Times Table Parkour**

Build a Parkour with one number. Make 2 rectangles for each way you can multiply it with same number (e.g., rectangles showing 3x7=21 and 7x3=21). |

When making planks and sticks from trees in Minecraft, students can explain how much of each item they need to craft other items. For example, it takes 2 planks to make 4 sticks. From there, they can make their own equations (Sticks=2*Planks) and go into ratios (2:1). |

**Multiply and Division Table Parkour**

Build a Parkour with one number. Make 1 rectangle to show multiplication separate that rectangle to show division. |

Build a Parkour with one number. Make 1 rectangle to show multiplication separate that rectangle to show division |

Students are using the equation P = 2(L+W) - 4 to find perimeter and A = L(W) to find area. |

Students are using multiplication and division to find the total number of materials needed to build a road in Minecraft. |

Students are using the formulas for perimeter and area to build their farm, which as addendum to their house. |

The Parthenon lesson plan involves multiplication using the formula for perimeter, area, and stairs. It also includes division when we figure out how many column blocks we need (C = H(P/2)). |

**Multiplication and Division Relationship**

Have students write and solve word problems using the formulas. L(W)=A by building rectangles in Minecraft. |

Have students write and solve word problems using the formulas. LxW=A by building rectangles in Minecraft |

When calculating the amount of planks and sticks they need for their tools, students will need to use multiplication and find the product. When determining the amount of wood needed, they will calculate the quotient. |

Students are figuring out the perimeter and area when the length and width have been determined. |

Students are using multiplication and division to determine the products and quotients of materials they need to build the road. |

Students use the formulas for area and perimeter when the length and width have been determined. |

After determining the length, width, and height of their structure, students will manipulate those numbers via multiplication and division to determine area, perimeter, blocks for stairs, and many other formulas. |

**Multiplication Walls**

Prompt students to build structures with equations like 5(x)=15 or 15/X=3. Combine with word problems. |

**Understand properties of multiplication and the relationship between multiplication and division.**CCSS.MATH.CONTENT.3.OA.B.5

In the Basic House, Farm, and Parthenon lesson plans, students learn are given opportunities to learn about the commutative, associative, and distributive properties of multiplication through the equations for perimeter and area. |

**Commutative Property of Multiplication**

3x5 = 15, 5x3 = 15 |

**Associative Property of Multiplication****Distributive Property of Multiplication**

This is the problem 4(3+2). The first image shows 3+2. The second image shows it being multiplied by 4. |

In the 'Building a Road' lesson plan, students become aware that the slabs they need to build are half of the blocks they will mine. Thus, dividing by 2 will show them the number of blocks they need. |

**Multiply and divide within 100.**CCSS.MATH.CONTENT.3.OA.C.7

This lesson involves students multiplying and dividing within 100, developing their knowledge of the relationship between multiplication and division. For example, students may say that they need 2 groups of 12 sticks, which equals 24 sticks total (2 x 12 = 24). |

**Solve problems involving the four operations, and identify and explain patterns in arithmetic.**CCSS.MATH.CONTENT.3.OA.D.8

Students will use addition, multiplication, and division in this lesson plan. They will also be exposed to remainders and the concept of rounding up. |

In the Algebra Architecture lesson plans, students are constantly using the four main operations with letters standing for unknown quantities. Students are using formulas to help them solve other formulas. |

**Use place value understanding and properties of operations to perform multi-digit arithmetic.**CCSS.MATH.CONTENT.3.NBT.A.1

**Base 10 - Place Value Structures**

423 |

423 is about 420 (rounding to nearest tens). |

420 is about 400 (rounding to nearest hundreds). |

**Base 10 - Expanded and Standard Forms**

This structure is an example of base 10. The blue blocks are the hundreds place, yellow blocks are the tens place, and grey blocks are the ones place. This is a model of 427. Expanded form: 400 + 20 + 7. You can have students go on the game, show them how to count base 10 numbers, have them create their own structures, and show their work via signs. |

**Base 10 - Regrouping**

This is an example of a base 10 structure that needs to be regrouped because there are 12 blocks in the tens place (yellow). When you put this structure in expanded form, it would be 300 + 120 + 7, which equals 427. Having students determine the value of the number and then regroup the blocks are essential skills in multiple digit addition and subtraction when carrying and borrowing are needed. |

This structure shows how the previous structure has been regrouped with the glass blocks indicating how the tens have been carried over to the hundreds. |

**Base 10 - Addition Algorithm**

This needs to be regrouped! This is a picture of 672 + 547. You add them together through regrouping. |

**Base 10 - Subtraction Algorithm**

1217-542=675 |

**Base 10 - Multiplication Algorithm**

This is the base 10 algorithm for multiplying with 1 digit to 2 digit numbers at this standard we will only multiply with multiples of 10. ‘ The is the Model for 80(5) Play close attention the numbers are set up in place value. In this picture the Stone blocks are in the 1’s and the lapps are 10’s Do you see how the 5 is directly behind the 0? That is not on accident. |

Step 1: Times the ones. 0 x 5 = 0 |

Step 2: The Ones Times the Tens. 5x80 = 400. but this can also been seen as 5x8 is 40 but just moved a place value over so is 400. |

**Develop understanding of fractions as numbers.**CCSS.MATH.CONTENT.3.NF.A.1

This farm has dimensions of 10 by 11 with a total area of 110 blocks. 38/110 are the stone blocks that make up the perimeter 64/110 are dirt blocks and 8/110 are water blocks 38/110+64/110+8/110=110/110=1 |

**Fraction Structures**

This image shows the problem of 1/3+1/3=2/3. |

This is an image of 3/5+1/5=4/5 |

- CCSS.MATH.CONTENT.3.NF.A.2
**Fractions on a Number Line**

- CCSS.MATH.CONTENT.3.NF.A.3.A
**Fractions on a Number Line**

This is a number line that is broken down into units of 1/3s. Meaning every 3 blocks is a whole number. The Lapis Block is setting on top of the number line at 1 1/3 or 4/3. |

- CCSS.MATH.CONTENT.3.NF.A.3.B
**Equivalent Fractions**

This image shows the fractions of 1/3, 2/6, 3/9, 4/12, 5/15. |

As you can see when you look at the structure this way it all looks like 1/3 |

In the lesson about making a farm, students are dealing with fractions that have the same denominator. In this example, the Perimeter = 38 and Area = 110. Perimeter/Area = 38/110, Dirt/Area = 64/110, and Water/Area =8/110. When the fractions are added together, it equals 110/110, which equals one farm. |

**Solve problems involving measurement and estimation.**CCSS.MATH.CONTENT.3.MD.A.1 CCSS.MATH.CONTENT.3.MD.A.2

**Represent and interpret data.**CCSS.MATH.CONTENT.3.MD.B.3

Before students make their farms in Minecraft, they will be using graph paper to sketch out what it will look like to ensure that their design will meet the design requirements. The blocks in Minecraft scale perfectly to the individual squares of the graph paper. |

**Geometric measurement: understand concepts of area and relate area to multiplication and to addition.**CCSS.MATH.CONTENT.3.MD.C.5

In the Algebra Architecture lesson plans which include building a house, farm, road, and Parthenon, students are measuring side lengths by blocks, which are essentially unit squares. |

In the Algebra Architecture lesson plans, students are measuring length and width to calculate the area of their structure, which can be measured in units squared. |

Students are using improvised units (e.g. blocks) to measure the area in the Algebra Architecture lesson plans. After calculating their area, they can check their work by counting the number of blocks. |

After measuring the length and width of their structure, students calculate the area by multiplying the dimensions. A = LxW. |

In the Algebra Architecture lesson plans, students are multiplying whole-number side lengths to calculate the area of their structures. |

Since structures in Minecraft are already decomposed into clearly visible units of measurement (blocks), it makes it very apparent that area is additive. |

**Geometric measurement: recognize perimeter.**CCSS.MATH.CONTENT.3.MD.D.8

In three of the five lessons of Algebra Architecture -- Building a House, Farm, and Parthenon -- students are solving for perimeter given the measurements of the dimensions of their structures. When given the opportunity to build multiple structures, students can see that different dimensions may have the same area but different perimeters (e.g. 8x5 and 10x4) and vice versa. |

**Reason with shapes and their attributes.**CCSS.MATH.CONTENT.3.G.A.1 CCSS.MATH.CONTENT.3.G.A.2

When constructing their farms, students can partition the area into equal parts (e.g. plots of dirt separated by water), and they are challenged to convert the area each component of the farm (perimeter, dirt, water) into a fraction, coming to a realization that they are part of a whole. |

## 4th Grade

- Operations & Algebraic Thinking
- Number & Operations in Base Ten
- Number & Operations - Fractions
- Measurement & Data
- Geometry

**Use the four operations with whole numbers to solve problems.**CCSS.MATH.CONTENT.4.OA.A.1

After walking through the process of creating the partial product structure, students should be able to set up the equation and generate the shape pattern for any multiplication equation. Furthermore, they should be able to recognize any erroneous calculations made (e.g. the dimensions of the structure not matching up to the numbers of the multiplication problem). |

This is additive comparison. Have students build answers to word problems. Mylo made 7 Minecraft videos and Jim made 4 more how many Minecraft videos did Jim make? |

This is multiplicative comparison. Mylo made 7 Minecraft videos and Jim made 4 times more Minecraft videos, how many Minecraft videos did Jim make? Students did the recognize and explain the difference between the 2 comparisons |

In the Crafting Calculator lesson plan, students will divide and deal with remainders when calculating how much wood they need to make their tools. Students will also realize they need to round up since it's not possible to collect a fraction of a wood block. |

In building their roads, students will have to divide whole numbers to calculate how many blocks they need to make their walkway. Depending on the length they choose, students may encounter remainders and round accordingly (S = 3L/2) |

**Gain familiarity with factors and multiples.**CCSS.MATH.CONTENT.4.OA.B.4

**Prime Factorization Trees**

Start from 54 (wood blocks). Its factors are 6 and 9 (birchwood) and the prime factors are shown in branch blocks. |

Associative property shown with this factor tree. |

**Generate and analyze patterns.**CCSS.MATH.CONTENT.4.OA.C.5

After walking through the process of creating the partial product structure, students should be able to set up the equation and generate the shape pattern for any multiplication equation. Furthermore, they should be able to recognize any erroneous calculations made (e.g. the dimensions of the structure not matching up to the numbers of the multiplication problem). |

In this problem students build towers starting at 1 and are give the rule +3 to the next tower, the numbers are 1, 4, 7, 10, 13 |

An other way to assess this is the have the students look at the number pattern 1, 4, 7, 10, 13 and find the next 3 numbers in the pattern and explain why. |

**Generalize place value understanding for multi-digit whole numbers.**CCSS.MATH.CONTENT.4.NBT.A.1

Using blocks to build place value structures, students develop an understanding that ten blocks in the ones place value equal that of one block in the tens place value, and so on. Students can look at these place value structures and interpret their numerical value. |

**Place Value Multiplication and Division**

Each Block in this model is worth 10. # There are 20 blocks adding to a total of 200. # They are divided into sections of 2 blocks add to a total of 20. # 200 divided by 20. |

**Use place value understanding and properties of operations to perform multi-digit arithmetic.**CCSS.MATH.CONTENT.4.NBT.B.4 CCSS.MATH.CONTENT.4.NBT.B.5

Creating the partial product structures by laying out the multi-digit numbers as dimensions is essentially an area model. In this lesson plan, students create the area model first (e.g. the answer) and then do some 'reverse engineering' through partial products to calculate the product. The area of the dimensions in the image above is also the answer to the multiplication problem, "16 x 15." |

**Extend understanding of fraction equivalence and ordering.**CCSS.MATH.CONTENT.4.NF.A.1 CCSS.MATH.CONTENT.4.NF.A.2

**Build fractions from unit fractions.**CCSS.MATH.CONTENT.4.NF.B.3

When sketching the designs of their farms, students must break down each of the components (dirt, water, perimeter enclosing) into fractions. |

After designing their farms, students are tasked with creating fractions for the perimeter, dirt, and water. In this example, the Perimeter = 40 and Area = 121. Perimeter/Area = 40/121, Dirt/Area = 64/121, and Water/Area =17/21. When the fractions are added together, it equals 121/121, which equals one farm. |

In designing their farm, students are turning parts of the whole into fractions and adding them back together, with like denominators. By building a digital farm, students are using a visual fraction model to learn about the concept. |

**Understand decimal notation for fractions, and compare decimal fractions.**CCSS.MATH.CONTENT.4.NF.C.5 CCSS.MATH.CONTENT.4.NF.C.6 CCSS.MATH.CONTENT.4.NF.C.7

**Solve problems involving measurement and conversion of measurements.**CCSS.MATH.CONTENT.4.MD.A.1 CCSS.MATH.CONTENT.4.MD.A.2 CCSS.MATH.CONTENT.4.MD.A.3

Students are directly using the formulas for area and perimeter to build digital structures, which can be applied to real-life contexts, such as architecture. |

**Represent and interpret data.**CCSS.MATH.CONTENT.4.MD.B.4

**Geometric measurement: understand concepts of angle and measure angles.**CCSS.MATH.CONTENT.4.MD.C.5 CCSS.MATH.CONTENT.4.MD.C.6 CCSS.MATH.CONTENT.4.MD.C.7

**Draw and identify lines and angles, and classify shapes by properties of their lines and angles.**CCSS.MATH.CONTENT.4.G.A.1 CCSS.MATH.CONTENT.4.G.A.2 CCSS.MATH.CONTENT.4.G.A.3

## 5th Grade

- Operations & Algebraic Thinking
- Number & Operations in Base Ten
- Number & Operations - Fractions
- Measurement & Data
- Geometry

**Write and interpret numerical expressions.**CCSS.MATH.CONTENT.5.OA.A.1

Students are using addition, multiplication, and division to solve the problem of how many materials they will need to craft their materials. They must calculate using the four operations and deal with remainders of quotients. |

In the house, farm, and Parthenon lesson plans, students will use various formulas, including area and perimeter, that contain parentheses; they must be able to evaluate these calculations correctly by applying the order of operations. |

In building the road, students will use formulas that have parenthesis and must be able to evaluate expressions with these symbols. |

When creating walls or columns, students will understand that finding the number of blocks involves manipulating the perimeter (multiplying and/or dividing the perimeter by a whole-number). By setting up the formula without solving it, students will be able to interpret the numeral expression before solving it. |

**Analyze patterns and relationships.**CCSS.MATH.CONTENT.5.OA.B.3

**Understand the place value system.**CCSS.MATH.CONTENT.5.NBT.A.1

When dealing with partial products, students will add them together using blocks. When the blocks in one place value are more than ten, they will carry that over to the place value to the left, showcasing an understanding of the relationships between place values. In the image above, students are using glass blocks to showcase their understanding that a block in one place represents ten blocks in the place to its right. |

**Expanded Forms with Decimals and Variables**

Have students be able to determine the value (221.95). |

Have students divide by place value. For example, divide the hundreds place by the tens place (200/20). |

**Perform operations with multi-digit whole numbers and with decimals to hundredths.**CCSS.MATH.CONTENT.5.NBT.B.5

**Partial Product Multiplication Structures****Place Value Multiplication Structures**Students are performing the standard algorithm for multiplication using partial products. Constructing the problems and building the place value structures in Minecraft allows them to check their calculations, and provides a visual representation of what is often taught as more of an abstract concept.

**Use equivalent fractions as a strategy to add and subtract fractions.**CCSS.MATH.CONTENT.5.NF.A.1 CCSS.MATH.CONTENT.5.NF.A.2

**Apply and extend previous understandings of multiplication and division.**CCSS.MATH.CONTENT.5.NF.B.3

When using the formula for slabs (S = 3L/2), students develop an understanding what each of the symbols and letters stand for. The 3 indicates the width of the road, L stands for a yet to be determined length, and the 2 shows that slabs are a half of a block. Thus, the student comes to an understanding that to get the number of blocks we need for the walkway of the road, we must multiply the length by the width and divide by 2. |

**Convert like measurement units within a given measurement system.**CCSS.MATH.CONTENT.5.MD.A.1

**Represent and interpret data.**CCSS.MATH.CONTENT.5.MD.B.2

**Geometric measurement: understand concepts of volume.**CCSS.MATH.CONTENT.5.MD.C.3 CCSS.MATH.CONTENT.5.MD.C.4 CCSS.MATH.CONTENT.5.MD.C.5

**Graph points on the coordinate plane to solve real-world and mathematical problems.**CCSS.MATH.CONTENT.5.G.A.1 CCSS.MATH.CONTENT.5.G.A.2

**Classify two-dimensional figures into categories based on their properties.**CCSS.MATH.CONTENT.5.G.B.3 CCSS.MATH.CONTENT.5.G.B.4

## 6th Grade

- Ratios & Proportional Relationships
- The Number System
- Expressions & Equations
- Geometry
- Statistics & Probability

**Understand ratio concepts and use ratio reasoning to solve problems.**CCSS.MATH.CONTENT.6.RP.A.1

Crafting the various tools in Minecraft involves different amounts of sticks and planks. The lesson plan Crafting Calculator uses this system to have students discuss these amounts in terms of ratios. For example, crafting a shovel needs two sticks and one plank, which is a ratio of 2:1. These numbers can grow more complex as students begin to calculate amounts for numerous tools (e.g. two shovels, a sword, and three axes). |

Since slabs are half of blocks but players can only mine in blocks, students must understand the ratio of blocks to slabs (1:2) to calculate how many blocks they need to build a slab walkway. |

**Apply and extend previous understandings of multiplication and division to divide fractions by fractions.**CCSS.MATH.CONTENT.6.NS.A.1

**Compute fluently with multi-digit numbers and find common factors and multiples.**CCSS.MATH.CONTENT.6.NS.B.2 CCSS.MATH.CONTENT.6.NS.B.3 CCSS.MATH.CONTENT.6.NS.B.4

**Apply and extend previous understandings of numbers to the system of rational numbers.**CCSS.MATH.CONTENT.6.NS.C.5 CCSS.MATH.CONTENT.6.NS.C.6 CCSS.MATH.CONTENT.6.NS.C.7

- CCSS.MATH.CONTENT.6.NS.C.7.A
- CCSS.MATH.CONTENT.6.NS.C.7.B
- CCSS.MATH.CONTENT.6.NS.C.7.C
- CCSS.MATH.CONTENT.6.NS.C.7.D

**Apply and extend previous understandings of arithmetic to algebraic expressions.**CCSS.MATH.CONTENT.6.EE.A.1 CCSS.MATH.CONTENT.6.EE.A.2

The equations for perimeter (P), area (A), and total number of blocks (Bh) for the basic house contain letters that stand for numbers and will change depending on values of other variables (length, width, height). |

In the Building the Road lesson plan, students must solve for the unknown variables of total number of blocks needed for the road (R), blocks for the border (B), and blocks for the walkway (C). |

In designing their farm, students must solve for the unknown variables of perimeter (P) and area (A). |

In the Parthenon lesson plan, students are solving for the unknown variables of total number of materials (M), stairs (S), total blocks (B), columns (C), area (A), and perimeter (P) when they determine what their values are for the length, width, and height. |

When solving the equation for perimeter in Minecraft, P = 2(L+W) - 4, students should be able to explain that perimeter is the "product of two times the sum of the length and width, minus four." They will do this throughout as the lessons challenge students to use language to describe each of the formulas being used. |

**Reason about and solve one-variable equations and inequalities.**CCSS.MATH.CONTENT.6.EE.B.5 CCSS.MATH.CONTENT.6.EE.B.6

The equations for the basic house, road, farm, and Parthenon lesson plans are comprised of perimeter, area, height, length, width - all mathematical terms that are used in real-life. Furthermore, students begin to build an understanding of how to read algebra as a language where variables stand for unknown numbers. |

In the house, farm, road, and Parthenon lesson plans, students must evaluate expressions in which letters stand for numbers on both sides of the equation (e.g., A = LxW), and build an understanding of the relationship between variables in a given expression. |

**Represent and analyze quantitative relationships between dependent and independent variables.**CCSS.MATH.CONTENT.6.EE.C.9

When making the five main tools in Minecraft, students will have to determine an algorithm to determine the number of wood, planks, and sticks needed to craft X amount of tools. |

Since many formulas used in the house, road, farm, and Parthenon lesson plans involve variables on both sides of the expression (A = LxW, P = 2(L+W) - 4, S = 3L/2), students will see that changing one variable will result in the change of the other variable. |

**Solve real-world and mathematical problems involving area, surface area, and volume.**CCSS.MATH.CONTENT.6.G.A.1

This farm is a polygon that is composed of decomposing shapes for example this farm has an area of 110 total blocks of the 110 blocks 38 are in the perimeter enclosure there are two 4 by 8 areas of dirt and 1 by 8 area of water. |

**Develop understanding of statistical variability.**CCSS.MATH.CONTENT.6.SP.A.1 CCSS.MATH.CONTENT.6.SP.A.2 CCSS.MATH.CONTENT.6.SP.A.3

**Summarize and describe distributions.**CCSS.MATH.CONTENT.6.SP.B.4 CCSS.MATH.CONTENT.6.SP.B.5

## 7th Grade

- Ratios & Proportional Relationships
- The Number System
- Expressions & Equations
- Geometry
- Statistics & Probability

**Analyze proportional relationships and use them to solve real-world and mathematical problems.**CCSS.MATH.CONTENT.7.RP.A.1 CCSS.MATH.CONTENT.7.RP.A.2

After understanding the ratios of blocks to slabs (1:2), students are using the formula S = 3L/2 to calculate how many blocks they need to craft the number of slabs required to make the walkway. |

**Apply and extend previous understandings of operations with fractions.**CCSS.MATH.CONTENT.7.NS.A.1

- CCSS.MATH.CONTENT.7.NS.A.1.A
- CCSS.MATH.CONTENT.7.NS.A.1.B
- CCSS.MATH.CONTENT.7.NS.A.1.C
- CCSS.MATH.CONTENT.7.NS.A.1.D

- CCSS.MATH.CONTENT.7.NS.A.2.A
- CCSS.MATH.CONTENT.7.NS.A.2.B
- CCSS.MATH.CONTENT.7.NS.A.2.C
- CCSS.MATH.CONTENT.7.NS.A.2.D

**Use properties of operations to generate equivalent expressions.**CCSS.MATH.CONTENT.7.EE.A.1 CCSS.MATH.CONTENT.7.EE.A.2

**Solve real-life and mathematical problems using numerical and algebraic expressions and equations.**CCSS.MATH.CONTENT.7.EE.B.3 CCSS.MATH.CONTENT.7.EE.B.4

In using the formulas for perimeter and area in the house, farm, and Parthenon lesson plans, students are manipulating the variables to determine their given values. For example, if a student wants to make their Area = 35 and are given width = 5, they must set up the equation to determine what the length would be. |

**Draw construct, and describe geometrical figures and describe the relationships between them.**CCSS.MATH.CONTENT.7.G.A.1 CCSS.MATH.CONTENT.7.G.A.2 CCSS.MATH.CONTENT.7.G.A.3

**Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.**CCSS.MATH.CONTENT.7.G.B.4 CCSS.MATH.CONTENT.7.G.B.5 CCSS.MATH.CONTENT.7.G.B.6

**Use random sampling to draw inferences about a population.**CCSS.MATH.CONTENT.7.SP.A.1 CCSS.MATH.CONTENT.7.SP.A.2

**Draw informal comparative inferences about two populations.**CCSS.MATH.CONTENT.7.SP.B.3 CCSS.MATH.CONTENT.7.SP.B.4

**Investigate chance processes and develop, use, and evaluate probability models.**CCSS.MATH.CONTENT.7.SP.C.5 CCSS.MATH.CONTENT.7.SP.C.6 CCSS.MATH.CONTENT.7.SP.C.7 CCSS.MATH.CONTENT.7.SP.C.8

## 8th Grade

**Know that there are numbers that are not rational, and approximate them by rational numbers.**CCSS.MATH.CONTENT.8.NS.A.1 CCSS.MATH.CONTENT.8.NS.A.2

**Expressions and Equations Work with radicals and integer exponents.**CCSS.MATH.CONTENT.8.EE.A.1 CCSS.MATH.CONTENT.8.EE.A.2 CCSS.MATH.CONTENT.8.EE.A.3 CCSS.MATH.CONTENT.8.EE.A.4

**Understand the connections between proportional relationships, lines, and linear equations.**CCSS.MATH.CONTENT.8.EE.B.5 CCSS.MATH.CONTENT.8.EE.B.6

**Analyze and solve linear equations and pairs of simultaneous linear equations.**CCSS.MATH.CONTENT.8.EE.C.7 CCSS.MATH.CONTENT.8.EE.C.8

**Define, evaluate, and compare functions.**CCSS.MATH.CONTENT.8.F.A.1 CCSS.MATH.CONTENT.8.F.A.2 CCSS.MATH.CONTENT.8.F.A.3

**Use functions to model relationships between quantities.**CCSS.MATH.CONTENT.8.F.B.4 CCSS.MATH.CONTENT.8.F.B.5

**Understand congruence and similarity using physical models, transparencies, or geometry software.**CCSS.MATH.CONTENT.8.G.A.1 CCSS.MATH.CONTENT.8.G.A.2 CCSS.MATH.CONTENT.8.G.A.3 CCSS.MATH.CONTENT.8.G.A.4 CCSS.MATH.CONTENT.8.G.A.5

**Understand and apply the Pythagorean Theorem.**CCSS.MATH.CONTENT.8.G.B.6 CCSS.MATH.CONTENT.8.G.B.7 CCSS.MATH.CONTENT.8.G.B.8

**Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.**CCSS.MATH.CONTENT.8.G.C.9

**Investigate patterns of association in bivariate data.**CCSS.MATH.CONTENT.8.SP.A.1 CCSS.MATH.CONTENT.8.SP.A.2 CCSS.MATH.CONTENT.8.SP.A.3 CCSS.MATH.CONTENT.8.SP.A.4