TEXAS SAMPLE PAGES STUDENT JOURNAL SENIOR AUTHORS PROGRAM CONSULTANTS. contributing authors. James Burnett Calvin Irons

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1 TEXAS PAGES SENIOR AUTHORS James Burnett Calvin Irons PROGRAM CONSULTANTS Diana Lambdin Frank Lester, Jr. Kit Norris contributing authors Debi DePaul Beth Lewis Peter Stowasser Allan Turton STUDENT JOURNAL

2 3.1 Step In Identifying Prime and Composite Numbers Color an array to represent each of these numbers What do you notice? What are some other prime numbers that you know? What are some composite numbers that you know? How could you prove that a number is composite? A composite number can be represented by an array that has more than one equal row. Step Up 13 A prime number is any whole number greater than zero that has exactly two unique factors itself and 1. A composite number is a whole number that has more than two whole number factors. 1. Color all the composite numbers. Then color a matching array to prove that each number is composite

3 Look at this chart. a. Loop the number 2. Draw a \ through all the multiples of 2. What do you notice? b. Loop the number 3. Draw a / through all the multiples of 3. What do you notice? c. Find the multiples of 6. What do you notice? d. Loop the number 5. Cross out all the multiples of 5. e. Loop the number 7. Cross out all the multiples of 7. f. Choose three numbers that have not been crossed out. What are their factors? a Step Ahead Write two prime numbers greater than

4 3.2 Step In Reviewing Multiplication Strategies Think about some of the different situations in which you use multiplication. Multiplication is often used to figure out the cost of purchases. Imagine you have to buy carpet for this floor area. Look at how these students figured out the area to be covered. Anna used partial products Nancy used a doubling-and-halving strategy is the same as yd Step Up Read these strategies for mentally calculating yd David used factors is the same as (3 4) 25 and (3 4) 25 is the same as 3 (4 25) Is there another way you could figure it out? Which way do you like best? Why? Use a strategy you like to calculate the area of a rectangle measuring 15 cm 24 cm. I multiplied 36 by 10. Then I multiplied my answer by 5 because 50 is 5 x 10. I multiplied 36 by 100. Then I halved my answer because 50 is one-half of

5 Use a method you like to calculate these. a = b = c = d = e = f = 2. Write how you could use the double-and-halve method to figure out Use the double-and-halve method to mentally calculate these. a = b = c = Step Ahead Write the missing numbers in each machine. a. IN OUT b. IN OUT

6 3.3 Step In Estimating to Solve Problems Involving Multiplication A coach is buying 12 of these shirts for his team. He has $400 to spend. Does he have enough money to buy the shirts? How could you estimate the total cost? $29 Two friends shared their strategies. Step Up Ashley used doubling and halving is the same as 6 60 I would round the price of one shirt to 30 first to make the estimation easier. Can you think of another way to make an estimate? How could you estimate the total cost of 15 caps? How did you round the amount to make your estimate? What strategy did you use to carry out the calculation? a. Buy 25. Diana used her understanding of place value is 36 so is Estimate the total cost. Use rounding to make the calculation easier. Show your thinking. $39 b. Buy 15. Baseball Ticket $11.98 $28 Estimate $ Estimate $ 60

7 Estimate the total cost. Use rounding to make the calculation easier. Show your thinking. a. Buy 20. $11.95 b. Buy 16. $25.12 Step Ahead Estimate $ 3. Use estimation to solve each problem. Show your thinking. a. A farmer is planting a field of lettuce. There are 45 rows which can each fit 68 plants. About how many seedlings will the farmer need to buy? Estimate $ b. A pet store has 25 fish tanks. 12 tanks each hold 26 fish and 13 tanks each hold 9 fish. About how many fish are there in total? seedlings Awan has $20. Estimate the number of each item he could buy. fish Ice Cream Soda Popcorn Meal Deal Ice Cream $3.95 Popcorn $4.98 Soda $3.50 Meal Deal $

8 3.4 Step In Using the Standard Algorithm to Multiply Three- and Two-Digit Numbers A ferry seats 132 people. It makes 24 trips each day. Does the ferry carry more or less than 2,500 people each day? 100 How could you figure out the exact number? Hugo drew this diagram to figure out the exact number. How will it help him? Write the partial products inside each part of the diagram. Nicole used the standard multiplication algorithm to calculate the total. How did she calculate the number in the first row? What does the red digit in the hundreds place represent? What numbers should she write in the second row? Write numbers to show your thinking. Then write the total. What is the total number of passengers that could travel on the ferry each day? Step Up 1. Write the partial product inside each part of the diagram below. Then add these to calculate the area =

9 Use the standard multiplication algorithm to calculate the exact product. Then estimate the product to check that your answer makes sense. a b c d. e Step Ahead Look at this calculation. Describe the mistake in words f

10 3.5 Step In Extending the Standard Multiplication Algorithm The local park is rectangular and measures 134 yd by 232 yd. How could you figure out the area of the park? Akeema drew this diagram of a rectangle split into parts to make it easier to multiply. 100 Write the partial product inside each part of her diagram. Add the partial products and write the area of the park below. Area is yd 2 Toby used the standard multiplication algorithm to calculate the area. What steps did he follow? Look carefully at the first and third row of his calculations. What do you notice? Why is the product in the third row 100 times greater than the product in the first row? Step Up Write the partial product inside each part of the diagram below. Then add these to calculate the area =

11 Use the standard multiplication algorithm to calculate the exact product. Then estimate the product to check that your answer makes sense. a b c d e f Step Ahead Color the beside the estimate that you think is closest to the exact product. a b. 7, c d. 6, ,000 12,000 42,000 6,000 13, ,000 4,200,000 60, , , ,000 1,200 13, , ,000 65

12 3.6 Step In Solving Word Problems Involving Multiplication (Large Numbers) This table shows the payments that players received after each game and the number of games that they played. How much did Player A earn this season? Step Up Player Payment Games Played A $4,350 4 B $1,025 5 C $ D $12, What number sentence would you write to show the problem? How could you calculate the total amount? Fatima used the standard algorithm to multiply like this How would you figure out the total amount that Player B received? 1. Look at the table above. Using the letter T for the unknown amount, complete a number sentence to show how to calculate the total amount paid to each of these players. Then figure out the total. Show your thinking. Player C T = T = Jude used a doubling strategy. Double 4,350 = 8,700 Double 8,700 = 17,400 Player D I would use the letter T for the unknown total. T = $4,350 4 $ $ 66

13 Solve these word problems. Show your thinking. a. It costs $795 to replace a backboard. 12 backboards were replaced in one season. What is the total cost of replacing them? b. It costs $7,320 to use the stadium for each game. What is the total cost to use the stadium for 41 games? c. There are 28 seats in each row. There are 42 rows. 2 seats in each row are reserved. What is the total number of seats available? $ seats d. Membership costs $245 for adults and $125 for children. There are 4,043 adult members. How much money has the club made from adult membership? $ $ Step Ahead Color the beside the number sentence that shows how to calculate the unknown total (T) in the following problem. Player E is paid $12,499 for each game played and a bonus of $10,000 for a season win. Player E participated in 42 games. His team won the season. What is the total amount Player E received? T = 42 $12,499 $10,000 T = $12,499 + $10, T = $12, $10,000 67

14 3.7 Step In Exploring Volume Place base-10 ones blocks on this base picture so it is six layers high. How can you figure out how many ones blocks you used? Complete this table to help you. What do you notice? Number of Cubes in Base Step Up a. Number of Layers 1. Place base-10 ones blocks on this base picture. Build up the number of layers to match the data in the table. Then complete the table. Number of Cubes in Base Total Number of Cubes How could you quickly figure out the total number of cubes in any object? What do you need to know? Number of Layers Total Number of Cubes The total number of cubes tells you the volume of the object. Volume is the amount of space that an object occupies

15 Complete these tables. You can use ones blocks to help. a. Number of Cubes in Base Number of Layers Total Number of Cubes b. c. Number of Cubes in Base Number of Cubes in Base Number of Layers 6 1 Number of Layers Total Number of Cubes Total Number of Cubes Step Ahead 1. Use 32 ones blocks to make an object that is the same on each layer. Draw the base of your object. 2. Write the missing numbers. a. Number of blocks in base b. Number of layers 69

16 3.8 Step In Analyzing Unit Cubes and Measuring Volume Jerilene was storing these boxes in the garage. How can she compare the amount of space that each box will occupy? To measure the space, she decides to fill each box with objects that are the same shape. How will this help? Look at these objects. Which object would you use to measure the volume of each box? How did you decide? Jerilene chose to use centimeter cubes to find the volume of the jewelery box. Does she need to fill the whole box with cubes? What is an easier way to figure out the volume? Just find the number of cubes in one layer. Then find the number of layers. Step Up 1. Use base-10 ones blocks to cover the area of this rectangle. Then complete the table. Dimensions of the Base of the Prism (cm) Number of Layers 1 Total Number of Centimeter Cubes

17 Use base-10 ones blocks to cover the area of this rectangle. Then complete the table. Dimensions of the Base of the Prism (cm) Number of Layers Total Number of Centimeter Cubes Complete each table to show the total number of centimeter cubes in each prism. Dimensions of the Base (cm) Number of Layers Total Number of Centimeter Cubes 3 5 Dimensions of the Base (cm) Number of Layers Total Number of Centimeter Cubes 4. Write a rule to figure out the total number of cubes in a prism when you know the dimensions of the base and the number of layers. Use your answers in Question 3 to help. Step Ahead Archie pours cubes into this container to figure out the volume. He counts 58 cubes. Do you think his calculation is accurate? Explain your thinking. 71

18 3.9 Step In Developing a Formula to Calculate Volume How can you figure out the volume of this prism without counting each individual cube? I know there are 8 cubes in the base. There are 4 layers = 32. Antonio multiplied the height of the prism by the number of cubes in the base. Base 8 cubes Height 4 layers 8 4 = 32 cubes Volume is 32 cubes. How are their methods similar? What rule could you write to match each method? Look at Kuma's method. Does it matter in what order she multiplies the dimensions? How do you know? Step Up 1. Imagine you built this prism with base-10 ones blocks. a. Complete this table. Length (Blocks) b. Write the volume of the prism. cm 3 2. Here are the dimensions of another prism. Width (Blocks) Height (Blocks) Length 8 cm Width 3 cm Height 5 cm Write how you can calculate the volume without counting blocks. Kuma multiplied the dimensions. Length 4 cubes Width 2 cubes = 32 cubes Volume is 32 cubes. Height 4 cubes Volume is usually measured in cubic units. The abbreviation for cubic centimeters is cm³. Total Number of Blocks 72

19 Use your rule from Question 2 to calculate the volume of these prisms. Length (cm) Width (cm) Height (cm) Volume (cm 3 ) a. b. c d. 4. Calculate the volume of each prism. Then write an equation to show the order that you multiplied the dimensions. a. b. c. Step Ahead cm 3 This square-based pyramid has been built with base-10 ones blocks. Calculate the volume of the pyramid. cm 3 cm 3 cm 3 73

20 3.10 Step In Finding the Dimensions of Prisms with a Given Volume The volume of a box is 60 in³. Write some possible dimensions for the box. = 60 in 3 = 60 in 3 = 60 in 3 How did you figure out the dimensions? What do you notice about each of the dimensions? How many different prisms can you make from a number that is prime? How do you know? Step Up Each dimension is a factor of has a lot of factors. 1. For each of these, draw and label the dimensions of a prism to match. Then write the volume. a. just less than 80 in 3 c. just less than 55 in 3 in 3 b. just more than 80 in 3 d. just more than 55 in 3 in 3 in 3 in 3 74

21 Complete each table to show the dimensions of four different prisms that have the same volume. a. Volume is 36 in 3 Length Width Height b. Volume is 64 in 3 Length Width Height c. Volume is 100 in 3 d. 3. Write the dimensions for another prism that has the same volume as 4 cm 8 cm 10 cm. Step Ahead Length Width Height Length cm Width cm Height cm Prism A is made with inch cubes. It is 4 cubes long, 5 cubes wide, and 2 cubes high. Prism B is made with centimeter cubes. It is 10 cubes long, 2 cubes wide, and 2 cubes high. Which prism has the greater volume? Explain your thinking. Volume is 72 in 3 Length Width Height 75

22 3.11 Step In Working with Volume The base of this prism has 6 sides. It is called a hexagonal-based prism. How could you calculate the volume of this prism? Andrea split the prism into two rectangular-based prisms. How will breaking the prism into parts help her figure out the volume? What number sentences would you write to match? Paul used a different strategy. He added more blocks to change the hexagonal-based prism into a rectangular-based prism. Step Up How could Paul's strategy help him figure out the volume of the prism? What number sentences would you write to match? 1. Each of these small cubes is 1 cm³. Figure out the volume of the prism. Write number sentences to show your thinking. cm 3 76

23 These prisms are made with centimeter cubes. Figure out the volume of each prism. Show your thinking. a. a. a. b. a. c. a. d. 3 cm 5 cm 8 cm 3 cm 3 cm cm 3 cm 3 e. 3 cm 2 cm 9 cm 4 cm cm 3 cm 3 5 cm cm 3 Step Ahead Some centimeter cubes have been removed from the middle of this prism. Figure out the volume of the new object. cm 3 77

24 3.12 Step In Solving Word Problems Involving Volume Emily is moving some household items into storage. She decides to pack the items into boxes. Boxes are sold in these three sizes. What is the volume of each box? How do you know? Write the volume on each box. Emily rents some storage space with the dimensions 10 ft 10 ft 8 ft. What is the volume of the storage space? She buys and fills 5 large boxes and 5 medium boxes. How much space do the boxes occupy in storage? How much storage space does she have left? Think about the dimensions of the boxes and the dimensions of the storage space. What size box would you use to fill the storage space? Why? Step Up 1 ft 2 ft 1 ft 1. Use the box sizes above. Figure out the total volume that each group of boxes would occupy. Show your thinking. a. 2 large boxes and 3 medium boxes 2 ft 3 ft 2 ft 3 ft Small Medium Large The height of the storage space is 8 feet so there would be some space left over if I used the medium boxes. b. 3 large boxes, 2 medium boxes and 6 small boxes 3 ft 4 ft ft 3 ft 3 78

25 Use the box sizes shown on page 78 to solve these word problems. Show your thinking. a. Helen buys and fills 4 boxes of each size. What is the total volume of the boxes? b. Jadyn has 5 medium boxes in the attic and 2 large boxes in the basement. Which group of boxes has the greater volume? Step Ahead ft 3 c. Ricardo has a storage space that measures 9 ft 9 ft 9 ft. What is the greatest number of medium boxes he can pack into this space? medium boxes Look at Question 2c above. After Ricardo packs the medium boxes into storage, how much space will be left over? ft 3 79