Section A Practice Problems

Problem 1

Here is a list of the first ten multiples of 5:

5, 10, 15, 20, 25, 30, 35, 40, 45, 50

  1. Circle the multiples of 10 in the list.

  2. What do you notice about where the multiples of 10 are on the list?

  3. Why do you think that is?

Show Solution
Solution
  1. 10, 20, 30, 40, and 50 are circled on the list.

  2. Sample responses:

    • They are the second, fourth, sixth, eighth, and tenth multiples of 5.
    • Every other multiple of 5 is a multiple of 10.
  3. Sample response: Every 2 groups of 5 is 1 group of 10, so a multiple of 10 would show up every second multiple of 5.

Problem 2

Find the value of each expression.

  1. 14×714 \times 7
  2. 13×613 \times 6
  3. 23×423 \times 4
  4. 85÷585 \div 5
Show Solution
Solution
  1. 98
  2. 78
  3. 92
  4. 17

Problem 3

There are 418 students at Jada’s school. There are 135 fewer students at Noah’s school. How many students are there at Jada’s and Noah’s schools together? Explain or show your reasoning.

Show Solution
Solution

There are 701 students at Jada’s and Noah’s schools. Sample responses:

418−135=283418 - 135 = 283

subtraction algorithm.

418+283=701418 + 283 = 701

addition algorithm.

Problem 4

  1. What is the value of the digit 6 in each of the numbers?

    • 165

    • 18,622

    • 675,219

  2. Complete this statement so that it is true:

    The value of the 6 in 675,219 is _______________ times that of the 6 in 165.

Show Solution
Solution
    • 60
    • 600
    • 600,000
  2. 10,000

Problem 5

Find the value of each sum and difference.

​​​​​

Show Solution
Solution

addition algorithm

subtraction algorithm

Problem 6

  1. Mai follows a rule to create a pattern of square blocks. Her rule is to keep adding 1 square to the top of her L design and 1 square to the right. Sketch or describe the next 2 shapes in Mai’s pattern.

    pattern of square blocks.
    pattern of square blocks. Step 1, 3 squares total, base of 2 squares, 1 square on top of the left square. Step 2, 5 squares total, base of 3 squares, 2 squares on top of the left square.

  2. Will Mai's pattern ever use 20 squares? Explain your reasoning.

Show Solution
Solution

  1. Sample response:

    pattern

  2. Sample response: No, the number of square pattern blocks is always odd.

Problem 7

Diego types the letters a, s, d, f and then repeats them in that order, over and over.

  1. What is the 5th letter Diego will type? What about the 10th? The 20th?

  2. Diego numbers each letter he types, starting with 1 for the first a. What are the numbers given to the first 6 f’s in his pattern? 

  3. What do you notice about the numbers for the f's?

Show Solution
Solution
  1. He'll type a for the 5th, s for the 10th, and f for the 20th.
  2. 4, 8, 12, 16, 20, 24

  3. Sample responses:

    • They are all multiples of 4.
    • They increase by 4 each time.

Problem 8

The rule for a pattern is “start with 8, keep adding 8.”

  1. Complete the table with the first 10 numbers in the pattern.

  2. What do you notice about the digits in the ones place? How do the digits change?

  3. Why do you think it changes that way?
start with 8,
keep adding 8

​​​​​

Show Solution
Solution
  1. 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
  2. Sample response: They start at 8, go down by 2 each time, and then repeat when it gets to 0 (8, 6, 4, 2, 0, 8, 6, 4, . . .).
  3. Sample response: Eight is 2 less than 10, so adding 8 is like adding 10 and then removing 2. Adding 10 to 8 gives 18, and removing 2 gives 16. Adding 10 to 16 gives 26, and removing 2 gives 24, and so on.

Problem 9

The rule for a pattern is “start with 25, keep adding 25.”

keep
adding 25		 25

  1. Complete the table with the first 8 numbers of the pattern.
  2. What do you notice about the numbers in the pattern? Explain or show why you think it happens.
  3. Could 475 be a number in this pattern? Explain or show your reasoning.
Show Solution
Solution
  1. 25, 50, 75, 100, 125, 150, 175, 200

  2. Sample responses:

    1. All the numbers are multiples of 25 because you are adding a group of 25 each time.
    2. The numbers go forth from odd to even because when you add 5 ones to a number with 5 ones, you make ten and ten is even. If you add 5 ones to a number that's a multiple of 10, you'll have 5 ones and 5 is odd.
    3. Every 4th number is a multiple of 100 because 25×4=10025 \times 4 = 100.
  3. Yes. Sample response: 475 is a multiple of 25. From the last number in the table, the pattern continues: 225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475.

Problem 10

  1. Make a list of the multiples of 2, 3, 4, 5, 6, 7, 8, 9, 10. Stop when you get a multiple of 10. For example, for 2, the list is 2, 4, 6, 8, 10.
  2. What do you notice about your lists? Make some observations.

Show Solution
Solution

  1. Lists of multiples:

    • 2, 4, 6, 8, 10
    • 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
    • 4, 8, 12, 16, 20
    • 5, 10
    • 6, 12, 18, 24, 30
    • 7, 14, 21, 28, 35, 42, 49, 56, 63, 70
    • 8, 16, 24, 32, 40
    • 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
    • 10

  2. Sample responses:

    • The lengths of the lists vary a lot. The last one is just 1 number long.
    • There is one that's 2 numbers long and then a bunch that are five numbers long and ten numbers long.
    • The ones that are five numbers long appear regularly.
    • Except for 10, it takes 5 multiples of even numbers to get to the first multiple of 10.
    • Except for 5, it takes 10 multiples of an odd number to get to the first multiple of 10.

Problem 11

Tyler draws this picture and writes the equation 1+3+5=91 + 3 + 5 = 9.

diagram. 3 rows of 3 circles of different colors. Top row, red, red, red. Middle row, blue, blue, red. Bottom row, black, blue, red.

  1. How do you think the equation relates to the picture?
  2. Tyler keeps drawing circles to make larger squares. How many new circles does he need to draw to make a 4-by-4 square, and then a 5-by-5 square?
  3. What pattern do you notice in the number of circles Tyler adds in each step?
  4. Why do you think the number of circles is increasing that way?

Show Solution
Solution
  1. There is one circle at the bottom left and then 3 highlighted with blue and 5 highlighted with red.
  2. He needs 7 more circles to make a 4-by-4 square and then 9 more circles to make a 5-by-5 square.
  3. Sample response: Tyler adds an odd number of circles each time. He started out with 1, and then added 3, 5, 7, 9, and so on.

  4. Sample response: To make a square that is 1 unit longer and wider, Tyler has to draw two equal groups of circles to the top and the right side, and then 1 in the upper-right corner. The two equal groups always make an even number. Adding 1 more makes it an odd number.

    diagram

Problem 12

Here is a growing pattern of squares that makes rectangles. The pattern follows the rule "keep adding 1 square to the row."

pattern of rectangles.
pattern of rectangles. Step 1, rectangle partitioned into 2 squares. Step 2, rectangle partitioned into 3 squares. Step 3, rectangle partitioned into 4 squares.

  1. Find the area and perimeter of the rectangles in steps 2 and 3.

    step number of squares area of rectangle (square units) perimeter of rectangle (units)
    1 2 2 6
    2
    3
  2. What do you notice when you look at the numbers in the chart? Use what you notice to complete the chart for steps 4 and 5.

  3. Draw the next two diagrams (for steps 4 and 5). Were your predictions for the area and perimeter of each rectangle correct?

  4. How would you describe what you noticed about this pattern to a classmate?

Show Solution
Solution
  1. See completed table.
  2. The first and second columns look like they go up by 1 each time and the third column goes up by 2. 
    step number of squares area of rectangle
    (square units)    		 perimeter of rectangle (units)
    1 2 2 6
    2 3 3 8
    3 4 4 10
    4 5 5 12
    5 6 6 14
  3. Drawings show a 5-by-1 rectangle and a 6-by-1 rectangle. Sample response: Yes, the area and perimeter in the drawings match the predictions.
  4. Sample response: The rectangle grows by 1 square each time. The rectangle’s area is always the same as the number of squares. The rectangle’s perimeter is always the number of squares times 2 plus 2.

Problem 13

Mai and Tyler make their own pattern. Mai's pattern repeats @, #, and $. Tyler's pattern repeats ~ and @.

Some of their pattern symbols are the same, some are different. The table shows the first 6 symbols in Mai’s pattern and the first 4 in Tyler’s pattern.

Mai's pattern @ # $ @ # $
Tyler's pattern ~ @ ~ @
  1. Complete the table with the next symbols in each pattern.
  2. At what step do you think Mai and Tyler will draw the same symbol at the same time? Explain how you know.

Show Solution
Solution
Sample responses:
  1. Mai's pattern @ # $ @ # $ @ #
    Tyler's pattern ~ @ ~ @ ~ @ ~ @
  2. 10th step. In Mai’s pattern, “@” is the 1st, 4th, and 7th symbol. Every third symbol after the first one is an “@” symbol. In Tyler’s pattern, it is the 2nd, 4th, 6th, and 8th symbol. Every other symbol is an “@.” If we extend the symbols, the 10th symbol will be an “@” in both patterns.