Panels 1 through 4 show a repeating shape pattern. What shape should appear in panel 5? State the repeating rule.

In this number wall, each brick above two others shows the sum of the bricks directly beneath it. What number belongs in the empty brick?

Fill in the missing number in the pattern and state the rule: \(5, 9, 13, \square, 21\).

This is a product wall. Each brick above two others shows the product of the bricks directly beneath it. Find the number in the empty brick.

These dot pictures grow in a pattern. How many dots should the next picture have? Explain the change.

In this addition wall, each brick shows the sum of the two bricks directly below it. Find the missing number in the bottom row.

A tile border begins: ○ □ □ ○ □ □ ○ □. What are the next two shapes? Which group repeats?

Find a simple rule that fits every shown top-to-bottom input-output pair. Then use that rule to complete the table. <table> <tr> <td>\(1200\)</td><td>\(5000\)</td><td>\(15{,}000\)</td><td>\(100{,}000\)</td><td>\(240{,}000\)</td><td>\(600{,}000\)</td> </tr> <tr> <td>\(3700\)</td><td>\(7500\)</td><td>\(17{,}500\)</td><td>\(102{,}500\)</td><td></td><td></td> </tr> </table>

Find a simple rule that fits every shown top-to-bottom input-output pair. Then use that rule to complete the table. <table> <tr> <td>\(15\)</td><td>\(60\)</td><td>\(200\)</td><td>\(1000\)</td><td>\(3000\)</td><td>\(12{,}000\)</td> </tr> <tr> <td>\(45\)</td><td>\(180\)</td><td>\(600\)</td><td>\(3000\)</td><td></td><td></td> </tr> </table>

Using a simple regular factor rule that fits the shown rows, continue the pattern with two more equations. Calculate every product, and describe how the products change. \(4\times 70=\square\) \(8\times 70=\square\) \(12\times 70=\square\) \(16\times 70=\square\)

Calculate each row. What pattern do you notice among the quotients within each row? a) \(240\div 4\), \(120\div 4\), \(60\div 4\) b) \(600\div 6\), \(300\div 6\), \(150\div 6\) c) \(800\div 8\), \(400\div 8\), \(200\div 8\)

Write the next division equation. Explain which number stays the same and how the other two change. \(24{,}000\div6=4000\) \(30{,}000\div6=5000\) \(36{,}000\div6=6000\) \(42{,}000\div6=7000\)

Give a simple regular rule that fits all the shown terms, then use that rule to continue the number pattern with three more numbers. \(125{,}400, 126{,}100, 126{,}800, 127{,}500, \ldots\)

Continue counting by \(500\) until you reach \(351{,}000\). \(348{,}000, 348{,}500, \ldots, 351{,}000\)

Find the step size and direction in each pattern. Then write the next three terms. a) \(456{,}700, 456{,}800, 456{,}900, \ldots\) b) \(820{,}000, 810{,}000, 800{,}000, \ldots\)

Count backward from \(1{,}000{,}000\) by \(1000\) until you reach \(994{,}000\). Write the complete sequence.

Fill in the missing numbers in this counting pattern. \(599{,}996, \square, \square, 599{,}999, \square, 600{,}001, \square\)

Find the rule and write the next two equations. \(120 \div 6 = 20\) \(240 \div 6 = 40\) \(360 \div 6 = 60\) ... ... How do the dividend and quotient change each step?

Count forward by \(100\) until you reach the next multiple of \(10{,}000\). a) \(359{,}400, 359{,}500, \ldots, 360{,}000\) b) \(819{,}600, 819{,}700, \ldots, 820{,}000\)

Count forward by ones. What three numbers come after \(399{,}999\)? \(399{,}997, 399{,}998, 399{,}999, \ldots\)

Multiply \(505\) by \(3\), \(5\), \(7\), and \(9\). 1. Calculate the four products. 2. Describe the digit pattern in the products.

Multiply \(198\) by \(2\), \(3\), \(4\), and \(5\). a) Calculate the four products. b) What do you notice about the tens digit of every product? c) Add the hundreds digit and ones digit of each product. What do you notice?

Continue the sequence using this rule: each new term is the sum of the two previous terms. Write the sequence through the sixth term. Starting terms: \(2400\) and \(3700\)

The terms shown are \(92, 84, 76, 68, 60, \ldots\). Find a simple repeating rule that fits these terms. Use your rule to write the next five terms.

A number pattern starts with \(2\) and \(3\). Each new term is the sum of the two terms immediately before it. Find the first ten terms.

A number pattern starts with \(10\). To find each new term, double the previous term and then subtract \(5\). Write the first six terms.

For each row, use a simple constant-step rule that fits all three shown terms, then write the next three terms. a) \(60{,}200, 60{,}100, 60{,}000, \ldots\) b) \(899{,}997, 899{,}998, 899{,}999, \ldots\) c) \(120{,}000, 110{,}000, 100{,}000, \ldots\)

Write the five numbers that come immediately before \(300{,}003\) in the counting sequence.

Count forward by tens. What four numbers come immediately after \(699{,}980\)?

Continue the number pattern. The first two terms are \(999{,}800\) and \(999{,}850\). Each term increases by the same amount. Write the next three terms.

Start at \(8970\) and list the next five numbers in each pattern. a) Count forward by tens. b) Count backward by hundreds. c) Count forward by fives.

Panels a), b), and c) show three rectangular dot arrays. Describe a rule relating the number of rows to the total number of dots. If the same rule is used for an array with \(9\) rows, how many dots will it contain?

The same triangular shape is shown in three successive stages. Describe the turn from one stage to the next. Which direction will the triangle point in Stage \(4\)?

The three panels show a growing square pattern on a unit grid. Each grid interval represents \(1\) unit. Describe how the side length changes. What will the side length and perimeter be in Stage \(4\)?

These are Stages \(1\) and \(2\) of a shape pattern. The same change is repeated at each step. a) What changes from Stage \(1\) to Stage \(2\)? Describe what happens to the point and the short slanted edge near the bottom. b) If you repeat that change, which way will the point face in Stage \(3\), and where will the short lower edge be?

Complete this addition wall. Each brick equals the sum of the two bricks directly beneath it. What is the top number?

In this addition wall, each brick equals the sum of the two bricks directly below it. Find the missing bottom number and the top number.

The rule has two steps: multiply by a whole number, then add a whole number. Use the same two numbers each time. <table><tr><th>Input</th><th>Output</th></tr><tr><td>1</td><td>4</td></tr><tr><td>2</td><td>7</td></tr><tr><td>3</td><td>10</td></tr><tr><td>5</td><td>\(\square\)</td></tr></table> Find the two numbers in the rule and the missing output.

Maya writes the pattern \(3,6,12,24\). She says the next number is \(27\). Is she right? Explain the rule and correct her prediction if needed.

Both addition walls follow the rule that the top brick is the sum of the bottom bricks. How much larger is the top brick in b) than in a)? Explain which change in the bottom row causes this.

Noor makes a straight row of touching squares from sticks. Each new square shares one side with the square before it. She has \(19\) sticks. Can she use every stick to make a row of complete squares? If so, how many squares?

The pattern is \(3,6,8,16,18,\dots\). What are the next two numbers? Explain how the pattern changes from one number to the next.

Two patterns start at \(2\). Pattern A adds \(2\) each time. Pattern B doubles each time. Find the number after two steps in each pattern. Why do the results differ even though both patterns start the same?

Wall a) is an addition wall: the top is the sum of the two bottom bricks. Wall b) is a product wall: the top is their product. Find both top numbers and explain why they differ.

Using a simple regular rule that fits all the shown terms, continue the sequence and explain that rule. \(320, 321, 323, 326, 330, \ldots, \ldots, \ldots, 356\)

Find a simple repeating rule that fits all the shown terms, then use that rule to fill in the missing terms. \(650, 660, 655, 665, 660, \ldots, \ldots, 675\)

Study this pattern of addition equations: \(245{,}100 + 312{,}400 = 557{,}500\) \(255{,}200 + 322{,}500 = 577{,}700\) \(265{,}300 + 332{,}600 = 597{,}900\) a) Continue the pattern by writing the fourth equation. b) By how much does the sum increase from one equation to the next? Explain using the addends.

One equation does not follow the pattern. 1) \(980{,}000 - 450{,}000 = 530{,}000\) 2) \(950{,}000 - 420{,}000 = 530{,}000\) 3) \(920{,}000 - 390{,}000 = 530{,}000\) 4) \(890{,}000 - 350{,}000 = 540{,}000\) a) Which equation breaks the pattern? b) Correct the fourth equation so that it follows the pattern.

Build an addition pattern using these rules: - The first equation is \(300{,}000 + 200{,}000 = 500{,}000\). - In each new equation, the first addend increases by \(50{,}000\). - In each new equation, the second addend decreases by \(20{,}000\). Find the fifth equation and its sum.

Choose two consecutive digits from \(1\) through \(9\), such as \(3\) and \(4\). a) Use the digits to form the two four-digit numbers with patterns \(abba\) and \(baab\). For \(3\) and \(4\), the numbers are \(3443\) and \(4334\). b) Subtract the smaller number from the larger number. c) Repeat with a different pair of consecutive digits. What pattern do you notice?

Investigate three-digit numbers and their reversed numbers. a) Choose a three-digit number that does not end in \(0\) and whose hundreds digit is exactly \(3\) greater than its ones digit. One example is \(653\). b) Reverse the digits and subtract the reversed number from your chosen number. c) Repeat with two other numbers that meet the same condition. What do you notice?

Use a simple regular rule for the two factors that fits the first four rows. One equation breaks that rule. Which equation is it? 1. \(10\times 100=1000\) 2. \(20\times 90=1800\) 3. \(30\times 80=2400\) 4. \(40\times 70=2800\) 5. \(50\times 50=2500\)

Using a simple regular rule that fits the shown factor changes, continue the multiplication pattern with three more equations and describe what happens to the products. \(10\times 90=\square\) \(20\times 80=\square\) \(30\times 70=\square\) \(40\times 60=\square\)

Calculate the quotients in this pattern. Then write the next two division equations and their answers. Explain how the divisors and quotients change. \(900\div9\) \(720\div8\) \(560\div7\) \(420\div6\) \(300\div5\)

Follow the pattern and write the next division equation. \(45\div8=5\,\text{R}\,5\) \(36\div8=4\,\text{R}\,4\) \(27\div8=3\,\text{R}\,3\) \(18\div8=2\,\text{R}\,2\) \(9\div8=1\,\text{R}\,1\)

Give a simple regular rule that fits all the shown division equations, then use it to write the next two equations. \(20\div2=10\) \(33\div3=11\) \(48\div4=12\) \(65\div5=13\) \(84\div6=14\)

Determine which division expressions have the same quotient. Make a prediction from the relationships among the numbers, explain it, and then verify each quotient. a) \(744\div 6\) b) \(372\div 3\) c) \(744\div 3\) d) \(248\div 2\)

Calculate and compare the quotients. How are the changes in the dividends related to each divisor? a) \(560\div 7\), \(567\div 7\), \(553\div 7\) b) \(420\div 6\), \(426\div 6\), \(414\div 6\) c) \(320\div 4\), \(328\div 4\), \(312\div 4\)

Calculate each group. What do you notice about the quotients? a) \(640\div 8\), \(320\div 4\), \(160\div 2\) b) \(420\div 6\), \(210\div 3\), \(70\div 1\) c) \(240\div 6\), \(120\div 3\), \(40\div 1\)

Write the next division equation and explain why its quotient will match the others. \(64{,}000\div8=8000\) \(56{,}000\div7=8000\) \(48{,}000\div6=8000\) \(40{,}000\div5=8000\)

Write the next division equation. Explain how you found all three numbers in it. \(21{,}000\div3=7000\) \(32{,}000\div4=8000\) \(45{,}000\div5=9000\) \(60{,}000\div6=10{,}000\)

Give a simple repeating rule that fits all the shown terms, then use that rule to continue the number pattern with three more numbers. \(850{,}000, 800{,}000, 825{,}000, 775{,}000, 800{,}000, \ldots\)

For each row, give a simple regular rule that fits all the shown terms, then use it to complete the number pattern. a) \(200{,}000, 350{,}000, 500{,}000, \square, \square, 950{,}000\) b) \(900{,}000, 820{,}000, 740{,}000, \square, \square, \square, 420{,}000\) c) \(125{,}000, 250{,}000, 500{,}000, \square\)

Give a simple regular rule that fits all the shown terms, then use it to fill the two blanks. \(100{,}000, 120{,}000, 150{,}000, \square, 240{,}000, 300{,}000, \square\)

Calculate and continue each sequence with three more expressions. Sequence 1: \(10 \times 20\), \(10 \times 30\), \(10 \times 40\), ... Sequence 2: \(5 \times 20\), \(5 \times 30\), \(5 \times 40\), ... Explain the pattern in the products of Sequence 1.

How many steps of the given size are needed to reach the next thousand? a) Start: \(342{,}400\); step size: \(200\) b) Start: \(712{,}750\); step size: \(50\) c) Start: \(899{,}920\); step size: \(20\)

Complete the table so that each row lands exactly on the target, which is the next thousand. <table><tr><th>Start</th><th>Step size</th><th>Target</th><th>Number of steps</th></tr><tr><td>\(451{,}200\)</td><td>\(200\)</td><td>\(452{,}000\)</td><td>?</td></tr><tr><td>\(123{,}750\)</td><td>?</td><td>\(124{,}000\)</td><td>\(5\)</td></tr><tr><td>?</td><td>\(10\)</td><td>\(888{,}000\)</td><td>\(3\)</td></tr></table>

In this number pattern, each step adds \(100{,}000 + 10{,}000 + 1000 + 100 + 10 + 1\). a) Continue with two more numbers: \(123{,}456 \to 234{,}567 \to \square \to \square\) b) Find the next number: \(489{,}000 \to 600{,}111 \to \square\) c) How many times must \(111{,}111\) be added to move from \(222{,}222\) to \(888{,}888\)?

Evaluate each set of division expressions. Then describe the pattern in the quotients. a) \(1{,}000{,}000 \div 1\) \(1{,}000{,}000 \div 10\) \(1{,}000{,}000 \div 100\) \(1{,}000{,}000 \div 1000\) b) \(100{,}000 \div 2\) \(100{,}000 \div 4\) \(100{,}000 \div 8\)

Find the step size in each pattern and write the next three terms. a) \(0, 150{,}000, 300{,}000, \ldots\) b) \(1{,}000{,}000, 925{,}000, 850{,}000, \ldots\) c) \(500{,}000, 575{,}000, 650{,}000, \ldots\)

Luke and Maya count within \(1{,}000{,}000\). Luke counts forward by \(250{,}000\), starting at \(0\). Maya counts backward by \(100{,}000\), starting at \(1{,}000{,}000\). Which numbers appear in both counting patterns?

Using a simple regular rule that fits the three shown equations, continue the pattern for two more rows and describe the sums. \(12{,}121+21{,}212=33{,}333\) \(23{,}232+21{,}212=44{,}444\) \(34{,}343+21{,}212=55{,}555\)

Using a simple regular rule that fits the three shown equations, write the next equation before calculating it. Then explain why all the shown and predicted sums are equal. \(12{,}345+87{,}654=99{,}999\) \(23{,}456+76{,}543=99{,}999\) \(34{,}567+65{,}432=99{,}999\)

Using a simple regular rule that fits the four shown equations, continue the pattern for two more rows and describe how the digits change. \(4321-2889=1432\) \(5432-2889=2543\) \(6543-2889=3654\) \(7654-2889=4765\)

Complete each set of equations. Then describe how the quotients change. a) \(48{,}000\div 2=\square\) \(48{,}000\div 4=\square\) \(48{,}000\div 6=\square\) \(48{,}000\div 8=\square\) b) \(12{,}000\div 3=\square\) \(24{,}000\div 3=\square\) \(36{,}000\div 3=\square\) \(48{,}000\div 3=\square\)

A number pattern starts at \(10\). Multiply in this repeating order: by \(10\), then by \(5\), then by \(2\). How many multiplication steps are needed to obtain a number greater than \(5000\)?

Using a simple regular rule that fits the shown equations, continue the sequence with one more equation, calculate all the quotients, and describe the pattern. \(1112\div 8\) \(2224\div 8\) \(3336\div 8\) \(\square\)

Study the three shown equations and use a simple regular rule that fits them. \(1\times9+2=11\) \(12\times9+3=111\) \(123\times9+4=1111\) a) Using that rule, write and evaluate a fourth equation. b) If the same rule is continued for a fifth row, predict the fifth result without carrying out a new multiplication. Explain how the result pattern supports your prediction.

Study the shown equations. Use a simple regular rule that fits them to answer both parts. \(6 + 6 = 12\) \(66 + 66 = 132\) \(666 + 666 = 1332\) \(6666 + 6666 = 13{,}332\) a) What is the next equation and result under that rule? b) Describe how the result changes when one more digit \(6\) is added to each addend.

Continue both number patterns logically, and find every sum. 1. \(210{,}000+320{,}000=\square\) 2. \(220{,}000+330{,}000=\square\) 3. \(230{,}000+340{,}000=\square\) 4. \(\square+\square=\square\) 5. \(\square+\square=\square\)

Choose two different digits from \(1\) through \(9\), and call them \(a\) and \(b\). a) Form the two four-digit palindromes \(abba\) and \(baab\). For example, the digits \(3\) and \(8\) form \(3883\) and \(8338\). b) Subtract the smaller palindrome from the larger one. c) Repeat the process with two other pairs of digits. d) Find the sum of the digits in each of your three differences. What do you notice?

Use two different digits to form the four-digit palindromes \(abba\) and \(baab\). a) Find the sums for the digit pairs \((2,5)\) and \((3,4)\). What do you notice? b) Which two different digits from \(1\) through \(9\) make the sum exactly \(8888\)? Find all three pairs.

A researcher multiplied \(606\) by several one-digit numbers and obtained \(1212\), \(1818\), \(2424\), and \(3030\). 1. Which one-digit factor produced each result? 2. Predict \(606\times 7\) using the pattern.

Multiply \(495\) by \(2\), \(3\), \(4\), and \(5\). a) Calculate the four products. b) Find the sum of the digits in each product. What do you notice? c) For each four-digit product, add the first and third digits, and then add the second and fourth digits. What do you notice?

A hundreds chart lists the numbers from \(1\) through \(100\) in ten rows. <table><tbody><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr><tr><td>11</td><td>12</td><td>13</td><td>14</td><td>15</td><td>16</td><td>17</td><td>18</td><td>19</td><td>20</td></tr><tr><td>21</td><td>22</td><td>23</td><td>24</td><td>25</td><td>26</td><td>27</td><td>28</td><td>29</td><td>30</td></tr><tr><td>31</td><td>32</td><td>33</td><td>34</td><td>35</td><td>36</td><td>37</td><td>38</td><td>39</td><td>40</td></tr><tr><td>41</td><td>42</td><td>43</td><td>44</td><td>45</td><td>46</td><td>47</td><td>48</td><td>49</td><td>50</td></tr><tr><td>51</td><td>52</td><td>53</td><td>54</td><td>55</td><td>56</td><td>57</td><td>58</td><td>59</td><td>60</td></tr><tr><td>61</td><td>62</td><td>63</td><td>64</td><td>65</td><td>66</td><td>67</td><td>68</td><td>69</td><td>70</td></tr><tr><td>71</td><td>72</td><td>73</td><td>74</td><td>75</td><td>76</td><td>77</td><td>78</td><td>79</td><td>80</td></tr><tr><td>81</td><td>82</td><td>83</td><td>84</td><td>85</td><td>86</td><td>87</td><td>88</td><td>89</td><td>90</td></tr><tr><td>91</td><td>92</td><td>93</td><td>94</td><td>95</td><td>96</td><td>97</td><td>98</td><td>99</td><td>100</td></tr></tbody></table> 1. Find \(1+100\). 2. Find \(2+99\). 3. Find \(3+98\). 4. What pattern do you notice? How many such pairs can be formed from all \(100\) numbers? 5. Use the pattern to find the sum of all the numbers from \(1\) through \(100\).

A four-digit repeating-block number has the form \(abab\). Swapping the digits gives \(baba\). a) Use the digits \(3\) and \(7\) to form \(3737\) and \(7373\). Subtract the smaller number from the larger number. b) Use two other digits that also differ by \(4\), such as \(1\) and \(5\). Find the new difference. What do you notice? c) Use the digits \(2\) and \(3\). How many times as large is the result from part a) as this new difference?

Investigate sums of four-digit repeating-block numbers of the forms \(abab\) and \(baba\). a) Use \(a=2\) and \(b=5\). Find \(2525+5252\). b) Use \(a=1\) and \(b=8\). Find \(1818+8181\). c) Yasmine claims, “If \(a+b=10\), then \(abab+baba\) is always \(11{,}110\).” Test the claim using the digit pairs \((3,7)\) and \((4,6)\).

Find the sum of all four-digit palindromes that begin and end with \(1\): \(1001,1111,1221,1331,1441,1551,1661,1771,1881,1991\)

In a sequence, each term is the sum of the two previous terms. The third term is \(12{,}500\), and the fourth term is \(20{,}000\). Find the first, second, and fifth terms.

Each sequence follows the rule that every new term is the sum of the two previous terms. The second term is always \(20{,}000\), while the first term increases by \(1000\) from one sequence to the next. Sequence 1: \(10{,}000, 20{,}000, \ldots\) Sequence 2: \(11{,}000, 20{,}000, \ldots\) Sequence 3: \(12{,}000, 20{,}000, \ldots\) Sequence 4: \(13{,}000, 20{,}000, \ldots\) Find the fifth term of each sequence. How do the fifth terms change?

Continue each sequence through the fifth term. Every new term is the sum of the two previous terms. The first term is always \(50{,}000\), while the second term increases by \(10{,}000\). a) \(50{,}000, 10{,}000, \ldots\) b) \(50{,}000, 20{,}000, \ldots\) c) \(50{,}000, 30{,}000, \ldots\) d) \(50{,}000, 40{,}000, \ldots\) By how much does the fifth term increase from one sequence to the next?

The terms shown are \(5, 15, 10, 30, 25, 75, \ldots\). Find a repeating two-step rule that fits these terms. Use your rule to write the next five terms.

Create a number pattern using these rules: - If the current number is odd, multiply it by \(3\) and add \(1\). - If the current number is even, divide it by \(2\). Start with \(13\). List every term until the pattern first reaches \(1\). How many operations are needed?

Create a number pattern using these rules: - If the number is less than \(20\), double it. - If the number is \(20\) or greater, subtract \(7\). Start with \(5\). Continue until a number appears for the second time. List the terms through that repetition. Which number repeats first?

A number machine uses these rules: - If a number ends in \(0\), remove the final zero. This is the same as dividing by \(10\). - If a number does not end in \(0\), add \(6\). a) Starting with \(14\), list the first eight terms. b) Describe the pattern you notice.

Study the multiplication equations. Use a simple regular rule that fits the shown equations for the continuation tasks. \(9 \times 7\) \(9 \times 77\) \(9 \times 777\) a) Calculate the three products. b) Describe the digit pattern in the products. c) Under the same simple rule, write the fifth equation and product without using the standard multiplication algorithm.

Study this pattern of sums: Row 1: \(1\) Row 2: \(1+2\) Row 3: \(1+2+3\) Row 4: \(1+2+3+4\) a) Find the results for rows \(1\) through \(6\). b) What is the result in row \(10\)? c) In which row is the result \(78\)? Explain how you found it.

A number machine multiplies each digit by itself and then adds the products. For example, \(13\) becomes \(1\times1+3\times3=10\), and \(10\) becomes \(1\times1+0\times0=1\). Start with \(49\). Repeat the rule until you reach \(1\). List the number chain and state how many steps it takes.

Use this number-machine rule: multiply each digit by itself and add the products. Start with \(20\) and apply the rule eight times. List the chain. What do you notice after the eighth step?

Investigate products with \(25\) and identify a pattern. a) Calculate: \(4 \times 25 = \square\) \(8 \times 25 = \square\) \(12 \times 25 = \square\) b) What happens to the product each time the first factor increases by \(4\)? c) Use the pattern from part b to find \(16 \times 25\) and \(20 \times 25\).

Complete the table. Starting from each number, count five steps in the stated direction. <table> <tr> <th>Starting number</th> <th>Direction</th> <th>Step size</th> <th>Step 1</th> <th>Step 2</th> <th>Step 3</th> <th>Step 4</th> <th>Step 5</th> </tr> <tr> <td>\(24{,}500\)</td> <td>forward</td> <td>\(250\)</td> <td></td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td>\(10{,}200\)</td> <td>backward</td> <td>\(40\)</td> <td></td> <td></td> <td></td> <td></td> <td></td> </tr> </table>

The two cube buildings occupy cubes along three perpendicular edges that meet at one corner of each outer box. a) In Building \(1\) and Building \(2\), how many cubes long is each occupied edge? b) A next building uses the same geometric rule inside a \(4 \times 4 \times 4\) box. How many cubes will it contain? Explain how you avoid counting the shared corner cube three times.

The diagrams show Lucas’s cube staircase on Days \(1\), \(2\), and \(3\). Describe the vertical stack heights in each shown staircase, listing them from tallest to shortest. If the same staircase-growth rule continues, what will the stack heights be on Day \(5\), and how many cubes will Day \(5\) contain altogether?

Some bricks are missing from these number walls. Every brick is the sum of the two bricks directly below it. Which walls can be completed using whole numbers? Select them and find the missing numbers.

Compare the two number walls. If every bottom-row value is doubled, what happens to the top value?

Complete the number wall. Then add \(2\) to every bottom-row value. By how much does the top value increase?

Fill the missing bottom bricks in the number wall shown. Every brick is the sum of the two bricks directly below it. Find one possible whole-number solution.

Jordan says, “If I increase the bottom-left brick of a four-brick number wall by \(10\), the top value also increases by exactly \(10\).” Is Jordan correct? Calculate wall b) to test the claim.

Complete both number walls and compare their top values. What happens when one inner bottom brick is increased by \(1\)?

The panels show a growing rectangle pattern on a unit grid. Each grid cell represents one square unit. Find the dimensions of Stage \(4\), and determine how many unit squares it contains. Explain the two changes that occur from one stage to the next.

Each stage adds one more square to a row of matchstick squares. Count the shared sides carefully. How many matchsticks will Stage \(5\) need? Explain what changes each time a square is added.

The three rectangles follow a pattern in both size and orientation. Each grid interval is \(1\) unit. Describe both rules, then state the dimensions and orientation of Stage \(4\).

The cube diagrams show square, one-layer stages of a growing pattern. If the pattern continues, how many cubes will Stage \(5\) contain? Explain how the side length and total number of cubes are related.

The diagrams show Stage \(1\) and Stage \(3\) of a one-layer cross pattern made from cubes. Candidates A and B are two possible missing Stage \(2\) shapes. Candidates A and B contain the same number of cubes. Which candidate continues the geometric pattern? Describe how each arm changes from Stage \(1\) to Stage \(2\) to Stage \(3\).

Stages \(1\), \(2\), and \(3\) show a shape pattern. Candidates A and B are possible next shapes. Which candidate continues the pattern? State the visual rule that determines your choice.

This sequence decreases. Give a simple regular rule that fits all the shown terms, then use it to fill in the three missing terms. \(880, 878, 874, 868, \ldots, \ldots, \ldots, 824\)

Give a simple repeating rule that fits all the shown terms, then use that rule to continue the number pattern with three more terms. \(2500, 5000, 4000, 8000, 7000, \ldots\)

A repeated operation adds \(1\), \(10\), \(100\), \(1000\), \(10{,}000\), and \(100{,}000\) to a starting number. a) Apply the complete operation twice to \(345{,}123\). What is the final number? b) Starting at \(100{,}000\), how many complete operations can you apply without going above \(1{,}000{,}000\)?

Using a simple regular rule that fits the three shown equations, continue the pattern for two more rows and explain the digit relationship you notice. \(10{,}203+19{,}998=30{,}201\) \(20{,}304+19{,}998=40{,}302\) \(30{,}405+19{,}998=50{,}403\)

Using a simple regular rule that fits the three shown equations, calculate the sums, write the next equation, and determine how much the sum increases each time. \(12{,}345+23{,}456\) \(23{,}456+34{,}567\) \(34{,}567+45{,}678\)

Study the three shown equations and give a simple regular rule that fits them. \(9\times9+7=88\) \(98\times9+6=888\) \(987\times9+5=8888\) a) Using that rule, write and evaluate the next equation. b) If the same rule is continued for one more row, predict the next result without carrying out a five-digit multiplication. How many digits would that result have?

Two four-digit palindromes of the forms \(abba\) and \(baab\), made from the same two digits, differ by exactly \(2673\). Which two digits could have been used? Find three different digit pairs.

A hundreds chart has ten numbers in each column. <table><tbody><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr><tr><td>11</td><td>12</td><td>13</td><td>14</td><td>15</td><td>16</td><td>17</td><td>18</td><td>19</td><td>20</td></tr><tr><td>21</td><td>22</td><td>23</td><td>24</td><td>25</td><td>26</td><td>27</td><td>28</td><td>29</td><td>30</td></tr><tr><td>31</td><td>32</td><td>33</td><td>34</td><td>35</td><td>36</td><td>37</td><td>38</td><td>39</td><td>40</td></tr><tr><td>41</td><td>42</td><td>43</td><td>44</td><td>45</td><td>46</td><td>47</td><td>48</td><td>49</td><td>50</td></tr><tr><td>51</td><td>52</td><td>53</td><td>54</td><td>55</td><td>56</td><td>57</td><td>58</td><td>59</td><td>60</td></tr><tr><td>61</td><td>62</td><td>63</td><td>64</td><td>65</td><td>66</td><td>67</td><td>68</td><td>69</td><td>70</td></tr><tr><td>71</td><td>72</td><td>73</td><td>74</td><td>75</td><td>76</td><td>77</td><td>78</td><td>79</td><td>80</td></tr><tr><td>81</td><td>82</td><td>83</td><td>84</td><td>85</td><td>86</td><td>87</td><td>88</td><td>89</td><td>90</td></tr><tr><td>91</td><td>92</td><td>93</td><td>94</td><td>95</td><td>96</td><td>97</td><td>98</td><td>99</td><td>100</td></tr></tbody></table> 1. Find the sum of the first column: \(1,11,21,31,41,51,61,71,81,91\). 2. Find the sum of the second column: \(2,12,22,32,42,52,62,72,82,92\). 3. How much greater is the second column sum? Explain why. 4. Find the sum of the tenth column, \(10,20,\dots,100\), without adding all ten numbers individually. 5. Add the ten column sums to find the total of the hundreds chart.

On a hundreds chart, the numbers go up by \(1\) as you move right and by \(10\) as you move down. This is a \(3\times3\) block from the chart: <table><tbody><tr><td>14</td><td>15</td><td>16</td></tr><tr><td>24</td><td>25</td><td>26</td></tr><tr><td>34</td><td>35</td><td>36</td></tr></tbody></table> a) Add its nine numbers. b) Multiply the center number by \(9\). What do you notice? c) Make a different \(3\times3\) block with \(58\) in the center. Does the same pattern work? d) Explain why this happens for any such block.

For four-digit repeating-block numbers of the forms \(abab\) and \(baba\), the difference depends on the difference between digits \(a\) and \(b\). a) Find the number difference when \(a=6\) and \(b=1\). b) Find every ordered digit pair \((a,b)\), with both digits from \(1\) through \(9\) and \(a>b\), for which the number difference is \(1818\). c) What is the smallest possible positive difference between two such numbers?

A sequence begins with \(6500\) and \(9000\). Each new term is the sum of the two previous terms. What is the first term greater than \(100{,}000\), and what position does it have in the sequence?

Two students build sequences in which each new term is the sum of the two previous terms. Starting with \(100{,}000\) and \(100{,}000\), the fifth term is \(500{,}000\). Now each student increases exactly one starting term by \(1\): Luke: \(100{,}001, 100{,}000, \ldots\) Mia: \(100{,}000, 100{,}001, \ldots\) Who gets the greater fifth term? Calculate both sequences to justify your answer.

The terms shown are \(2, 3, 6, 11, 18, \ldots\). Find a simple rule for the changes between terms that fits the sequence. Use that rule to find the next five terms.

In a number pattern, each term is the sum of the two terms before it. Two consecutive terms are \(7\) and \(11\). Find the two terms immediately before \(7\) and the two terms immediately after \(11\).

Leo and a friend play “Reach \(32\).” They start at \(0\) and take turns adding \(1\), \(2\), \(3\), \(4\), or \(5\) to the total. The player who reaches exactly \(32\) wins. a) Leo goes first and adds \(2\). Why is that a good first move? b) What should Leo add if his friend adds \(1\)? What if his friend adds \(4\)? Explain how Leo can answer any allowed move. c) What totals should Leo reach after his turns to make sure he wins?

Two players take turns taking \(1\) or \(2\) pieces of candy from a bowl holding \(21\) pieces. The player who takes the last piece wins. If both play as well as they can, which player can make sure of winning? Explain the moves.

Study this sequence of expressions: 1. \(2-1\) 2. \(4-3+2-1\) 3. \(6-5+4-3+2-1\) 4. \(8-7+6-5+4-3+2-1\) a) Find the results of the first four rows. Describe the pattern. b) Write the expression for row \(6\) and find its result. c) Which row has a result of \(25\)?

Study this sequence of calculations: 1. \(100-1=99\) 2. \(100-(1+2)=97\) 3. \(100-(1+2+3)=94\) 4. \(100-(1+2+3+4)=90\) a) Find the results for rows \(5\) and \(6\). b) In which row is the result less than \(80\) for the first time? c) What is the result in row \(10\)?

A “happy number” reaches \(1\) after repeatedly multiplying each digit by itself and adding the products. Test \(13\), \(14\), and \(15\). Which are happy numbers? Show each chain until it reaches \(1\) or repeats a number.

A number pattern starts at \(15{,}000\). After five equal steps forward, it reaches \(17{,}500\). a) Find the size of one step. b) List the five numbers reached by moving forward. c) List the five numbers reached by moving backward from \(15{,}000\) using the same step size.

Zoe and Ben each build a number wall. Every brick is the sum of the two bricks directly below it. The two walls are shown. a) Find the top brick in Zoe's wall a). b) Find the top brick in Ben's wall b). c) How much greater is Ben's top brick? Explain why the one changed bottom brick creates this exact change at the top.

Mia and Jonas compare the two number walls shown. Every brick is the sum of the two bricks directly below it. Jonas claims, “Because the changed bottom brick increased by \(10\), the top brick will also increase by exactly \(10\).” Is Jonas correct? Find the top brick in each wall and explain your answer.

Complete both large number walls. Every brick is the sum of the two bricks directly below it.

Complete both number walls. Decide where to begin in each wall.

This number wall has five rows. Each brick is the sum of the two bricks below it. Find every missing value, and use the symmetry you notice to help you.

Complete the five-row number wall. The values are large, but the rule is unchanged: each brick is the sum of the two bricks below it.

The top value must become \(800\). Add the same number to each bottom-row value. What will the new bottom row be?

Subtract the same amount from each bottom-row value so that the top becomes exactly \(120\). What is the new bottom row?

What happens to the top value of a four-brick number wall when the second bottom brick is increased by \(10\)? Compare the two walls and explain why.

Complete the number wall to the top. Each brick is the sum of the two bricks below it. Use the mirror symmetry in the bottom row.

Each brick is the sum of the two bricks below it. The number wall is symmetric. Work backward to complete it.

Complete the number wall and describe the pattern in the bottom row.

Compare the three product walls. In a product wall, each brick is the product of the two bricks directly below it. Find each top value. Which walls have the same result, and why is wall c) different?