Section 1
Looking for Patterns
Here are the first few rows of Pascal's triangle:a. How many numbers are in each row shown?b. How many numbers are in Row $4 ?$ In Row $5 ?$ In Row $6 ?$c. If you are given a row number, how can you determine how many numbers are in that row?d. In some rows, every number appears twice. Other rows have a middle number that appears only once. Will Row 10 have a middle number? Will Row $9 ?$ How do you know?
A certain row of Pascal's triangle has 252 as the middle number and 210 just to the right of the middle number.a. What is the number just to the left of the middle number? How do you know?b. What is the middle number two rows later? How do you know?
Describe the pattern in each sequence, and use the pattern to find the next three terms.3, 12, 48, 192, __ , __ , __
Describe the pattern in each sequence, and use the pattern to find the next three terms.0.1, 0.4, 0.7, 1.0, __ , __ , __
Describe the pattern in each sequence, and use the pattern to find the next three terms.2, 5, 4, 7, 6, 9, __, __, __
Describe the pattern in each sequence, and use the pattern to find the next three terms.$\Delta, \infty, \Delta, \Delta, \infty, \Delta, \Delta, \Delta, \infty$ __, __, __
Describe the pattern in each sequence, and use the pattern to find the next three terms.-5,-4,-3,-2, __ , __, __
Describe the pattern in each sequence, and use the pattern to find the next three terms.a, c, e, g, __, __, __
Some patterns in Pascal's triangle appear in unexpected ways. For example, look at the pattern in the sums of the rows.a. Find the sum of each row shown above.b. Describe the pattern in the row sums.
The pattern below involves two rows of numbers. If the pattern were continued, what number would be directly to the right of $98 ?$ Explain how you know.$$\begin{array}{cccccccccc} & 3 & & 6 & & 9 & & 12 & & 15 & & 18 \\1 & 2 & 4 & 5 & 7 & 8 & 10 & 11 & 13 & 14 & 16 & 17\end{array}$$
Look at this pattern of numbers. If it were continued, what number would be directly below $100 ?$$$\begin{array}{ccccc} & & 1 & & \\& 2 & 3 & 4 & \\5 & 6 & 7 & 8 & 9 \\10 & 11 & 12 & 13 & 14 & 15\end{array}$$
For this problem, you may want to draw the shapes on graph paper.a. Find the next term in this sequence:b. This table shows the number of squares in the bottom rows of Terms 1 and $2 .$ Copy and complete the table to show the number of squares in the bottom rows of the next two terms.$$\begin{array}{|c|c|}\hline \text { Term } & \begin{array}{c}\text { Squares in } \\\text { Bottom Row }\end{array} \\\hline 1 & 1 \\\hline 2 & 3 \\\hline 3 & \\\hline 4 & \\\hline\end{array}$$c. Look at your table carefully. Describe the pattern of numbers in the second column. Use your pattern to extend the table to show the number of squares in the bottom rows of Terms 5 and 6d. Predict the number of squares in the bottom row of Term 30e. Now make a table to show the total number of squares in each of the first five terms.$$\begin{array}{|c|c|}\hline \text { Term } & \text { Total Number of Squares } \\\hline 1 & 1 \\\hline 2 & 4 \\\hline 3 & \\\hline 4 & \\\hline 5 & \\\hline\end{array}$$f. Look for a pattern in your table from Part e. Use the pattern to predict the total number of squares in Term 10
Imagine that an ant is standing in the square labeled $A$ on the grid below. The ant can move horizontally or vertically, with each step taking him one square from where he started.a. On a copy of the grid, color each square (except the center square) according to the least number of steps it takes the ant to get there. Use one color for all squares that are one step away, another color for all squares that are two steps away, and so on.b. What shapes are formed by squares of the same color? How many squares of each color are there? What other patterns do you notice?
Find each sum or difference without using a calculator.$5,853-788$
Find each sum or difference without using a calculator.$1,054+1,492$
Find each sum or difference without using a calculator.$47,745-2,943$
Write thirty-two thousand, five hundred sixty-three in standard form.
Write fourteen million, three hundred two thousand, two in standard form.
Write 324 in words.
Write 12,640 in words.
Imagine that you have 12 square tiles, each measuring 1 inch on a side. a. In how many different ways can you put all 12 tiles together to make a rectangle? Sketch each possible rectangle.b. Which of your rectangles has the greatest perimeter? What is its perimeter?c. Which of your rectangles has the least perimeter? What is its perimeter?