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**Jacob’s Birthday Party**

Jacob’s family was setting up tables for his birthday party. They didn’t want any of the kids to feel left out, so they decided to push all of the small tables together to make one long table for everyone. How many tables does Jacob need for 54 guests?

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**How do each of these rules make sense geometrically?**

y=4x+2 y=6x-2(x-1) y=5+4(x-2)+5 y=6+4(x-1) y=6x-2(x-2)-2 y=5x-(x-2)

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I am having a party, and I want to give 2 Peeps to each of my guests as part of a party favor bag. Because the Peep rush has ended, Haribo is having a special promotion. For each package of Peeps that you order, you get 2 Peeps as a gift. How many packages of Peeps should I order for my 84 guests? How many Peeps will I be paying for and how many will be free?

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Stage 1 Stage Stage 3

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Stage Stage Stage 3

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Stage Stage Stage 3 2 + (n+1) 3 + n

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All of the bridges in this part are built with yellow rods for spans and red rods for supports, like the ones shown here. This is a 2-span bridge. Note that the yellow rods are 5 cm long. Now build a 3-span bridge. How many yellow rods did you use? How long is your bridge? How many red rods did you use? How many rods did you use altogether? Try to answer the questions without building a 5-span bridge. How many yellow rods would you need for a 5-span bridge? How long would your bridge be? How many red rods would you need? How many rods would you need altogether? Write a rule for figuring out the total number of rods you would need to build a bridge if you knew how many spans the bridge had.

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In the diagram below, the shaded hexagons are flower beds, and the white hexagons are white paving stones. Marco figures out how many white paving stones are needed around different numbers of flower beds. Draw a diagram to show how many white stones are needed around 5 flower beds. Marco says that 28 white stones are needed around 13 flower beds. Without drawing the flower beds, explain how you know that Marco is not correct. How many white stones are needed around 13 flower beds?

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**Tom uses toothpicks to make the shapes in the diagram below.**

How many toothpicks make shape 3? Draw shape 4 next to the diagram above. Tom says, “I need 36 toothpicks to make shape 12.” Tom is not correct. Explain why he is not correct. How many toothpicks are needed to make shape 12?

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**How many seats fit around a row of triangular tables?**

INPUT RULE OUTPUT Number of Δ tables Number of Seats What patterns do you see? INPUT (Number of tables) OUTPUT (Number of seats)

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**If you know how many degrees are in a triangle, how many are in this figure?**

Complete the table. What do you notice? Write a rule to describe the pattern.

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Blocks are used to build the staircases shown below (1 for the first, 3 for the second, 6 for the third, etc). How many blocks will be used for the 100th stair? Can you generate a rule that could be used to find the number of blocks used for any number of stairs? You could also say these are toothpicks and ask how many toothpicks would be required. Rather than a formula of N(n+1) or (n^2 + n)/2 you would get a formula of something that is equivalent to x^2 + 3x (I originally got (x+1)^2 + (x-1) 2 OR you could ask for the perimeter and get the formula of 4n

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**How many blocks would be required to build 100 stairs?**

What is a general rule that could be used to find the number of blocks to build any number of stairs? 1 Stair 2 Stairs 3 Stair

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**Describe a pattern you see in the cube buildings.**

Use your pattern to write an expression for the number of cubes in the nth building, where n is an integer. Use your expression to find the number of cubes in the fifth building.

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4 cows 5 cows 6 cows The pens above show how much fencing is required to contain 4, 5, and 6 cows. Each square represents 1 length of fence. How many lengths would be required to contain 100 cows? What is a general rule that could be used to determine how many lengths are required for any number of cows

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Case Case Case 3 How is the above pattern growing? Represent the pattern in a t-table, a graph, in words, and with an algebraic rule. How many squares would be in Case 100?

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Term Term 2 Draw Term 3. Create a table, graph, and equation to represent the pattern you see. How many squares would be used to build Term 50?

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Term Term 2 Draw Term 3. Create a table, graph, and equation to represent the pattern you see. How many squares would be used to build Term 50? Does the shading help you to see the pattern?

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**Below is a 10x10 grid with the border shaded in**

Below is a 10x10 grid with the border shaded in. Generate a rule that could be used to determine the number of shaded border squares for any square grid

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The numbers of dots in the figures below are the first four rectangular numbers. Assume the the pattern continues. Write down everything you observe about the patter. What are the first four rectangular numbers? How do the numeric values relate to the picture? Describe the picture of the 10th rectangular number? Use words, diagrams, or symbols to generalize the pattern. How do you know your generalization is true?

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**Find the number of cubes in the tenth tower. **

How many cubes would be in the Zero-th building? What is the rate of change? How could the pattern be described symbolically? Or, if you know the pattern number, can you write a formula that will give you the height of any tower? #1 #2 #3 #4

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**Find the number of cubes in the tenth tower. **

How many cubes would be in the Zero-th building? What is the rate of change? How could the pattern be described symbolically? Or, if you know the pattern number, can you write a formula that will give you the height of any tower? #1 #2 #3 #4

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**Find the number of cubes in the tenth tower. **

How many cubes would be in the Zero-th building? (Think of it as underground/in the basement) What is the rate of change? How could the pattern be described symbolically? Or, if you know the pattern number, can you write a formula that will give you the height of any tower? #1 #2 #3 #4

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**Find the number of cubes in the tenth tower. **

How many cubes would be in the Zero-th building? (Think of it as underground/in the basement) How could the pattern be described symbolically? Or, if you know the pattern number, can you write a formula that will give you the height of any tower? #1 #2 #3 #4 Discuss this question in your small group. Be prepared to share your ideas in a large group discussion. What is the secret to finding the growth rates and equations for the stacking cube patterns?

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**How many toothpicks will you need for 4 sections? 5 sections?**

What patterns do you see? Explain. How about 30 sections? sections? How did you decide? Explain.

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**How many toothpicks will you need for a square with a side length 4? 5?**

What patterns do you see? Explain. How about a side length of 20? Side length 100? How did you decide? Explain.

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**How many toothpicks will you need for 4 squares? 5 squares?**

What patterns do you see? Explain. How about 30 squares? squares? How did you decide? Explain.

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**How many toothpicks will you need for 4 wiggles? 5 wiggles?**

What patterns do you see? Explain. How about 30 wiggles? wiggles? How did you decide? Explain.

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**How many square will you need for figure 4? Figure 5?**

What patterns do you see? Explain. How about figure 10? Figure 30? How did you decide? Explain.

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**How many sides will show in a 4-long? 5-long?**

What patterns do you see? Explain. How about a 20-long? long? How did you decide? Explain.

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**How many triangles will you make for 4-high? 5-high?**

What patterns do you see? Explain. How about a 30-high? high? How did you decide? Explain.

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**How many squares will you need for Figure 4? Figure 5?**

What patterns do you see? Explain. How about Figure 20? Figure 50? How did you decide? Explain.

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**How many squares will you need for Figure 4? Figure 5?**

What patterns do you see? Explain. How about Figure 20? Figure 50? How did you decide? Explain.

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**How many squares will you need for Figure 4? Figure 5?**

What patterns do you see? Explain. How about Figure 10? Figure 30? How did you decide? Explain.

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**How many squares will you need for Figure 4? Figure 5?**

What patterns do you see? Explain. How about Figure 10? Figure 30? How did you decide? Explain.

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**How many squares will you need for Figure 4? Figure 5?**

What patterns do you see? Explain. How about Figure 10? Figure 30? How did you decide? Explain.

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**Square tiles were used to make the pattern above.**

a. Write an equation for number of tiles needed to make the nth figure. Explain. b. Find an equivalent expression for the number of tiles in part a. Explain why they are equivalent. c. Write an equation for the perimeter of the nth figure. d. Identify and describe the figure in the pattern that can be made with exactly 420 tiles. e. Describe the relationship represented by the equation in parts a and c.

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Draw the next figure. Explain how you knew what figure would come next. What is the pattern that you see? How many circles would be in the next term? How many circles would be in the 25th term? How do you know? What is a general rule that would allow you to find the number of circles in any term?

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Mrs. Ramirez is tiling her back walk and has designed her own hexagon mosaic tiles. They consist of: trapezoids, rhombi, and triangles. The perimeter of the space she has outlined is 106 ft., and she only wants one tile across.

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**Each mosaic tile contains:**

6 trapezoids 2 rhombi 2 triangles The cost for each individual tile is: Trapezoids - $3/tile Rhombi - $2/tile Triangles - $1/tile How much will it cost to tile her back walk?

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