Math News! 4 th Grade Math

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1 Grade 4, Module, Topic A Module : Unit Conversions and Problem Solving with Metric Measurement better understanding of the math concepts found in Eureka Math ( 013 Common Core, Inc.) that is also posted as the Engage New York material which is taught in the classroom. Module of Eureka Math (Engage New York) covers Unit Conversions and Problem Solving with Metric Measurement. Focus Area Topic A: Metric Unit Conversions Converting Units Students review place value concepts while building fluency to decompose or convert from larger to smaller units. They learn 1 meter (m) is equal to 100 centimeters (cm) just as 1 hundred is equal to 100 ones. The table below continues this thinking. Focus Area Topic A: Metric Unit Conversions Words to Know: Measurement quantity as determined by comparison with a standard unit Convert - to express a measurement in a different unit Distance -the length of the line segment joining two points Students use this knowledge of unit conversion to solve addition and subtraction problems involving mixed units. Example Problem and Answer Mixed Unit refers to numbers that are paired but represent individual entities or units. Take the number 35. Written this way, it represents 35 ones. That is 35 of the same unit. However, if we write 3 tens 5 ones, then we have mixed units because the 3 and the 5 are conveying different meanings. OBJECTIVES OF TOPIC A Express metric length measurements in terms of a smaller unit; model and solve addition and subtraction word problems involving metric length. Express metric mass measurements in terms of a smaller unit; model and solve addition and subtraction word problems involving metric mass. Express metric capacity measurements in terms of a smaller unit; model and solve addition and subtraction word problems involving metric capacity. In this example, the units are added separately. The meters are added together and the centimeters are added together. Then the centimeters are converted to meters. Can you think of another way to solve it?

2 Focus Area Topic A: Metric Unit Conversions Converting Units Conversions between the units are recorded in a two-column table. Recording the unit conversions in a table allows students to see the ease of converting from a smaller unit to a larger unit. Module : Unit Conversions and Problem Solving with Metric Measurement Strategies for Adding and Subtracting Mixed Units Addition and subtraction single-step problems of metric units provide an opportunity to practice using simplifying strategies as well as solve using the addition and subtraction algorithm. In this next example, the student uses the simplifying strategy. Example Problem and Answer Jon had a cooler that had 3L 40mL of water in it. He emptied a container with 13L 585mL of water into the cooler. How much water is in the cooler now? Strategies for Adding and Subtracting Mixed Units Students will be taught several different strategies for adding and subtracting mixed units. In the following example, the student will use an algorithm strategy and decompose or convert the kilograms to grams before solving. Example Problem and Answer Together, a squirrel and a beaver weigh 6 kg 30 g. If the squirrel weighs 1 kg 55 g, how much does the beaver weigh?

3 Grade 4, Module 3, Topic B Math ( 013 Common Core, Inc.) that is also posted as the newsletter will discuss Module 3, Topic B. B. Multiplication by 10, 100, and 1,000 Words to know Area Model Number Disk Place Value Chart Bundle Helpful Hints!!! ones x tens = tens tens x tens = hundreds hundreds x tens = thousands 40 x 10 = 4 tens x 1 tens 40 x 100 = 40 x 10 x 10 = 4 tens x 1 ten x 1 ten Decompose separate numbers into smaller numbers 40 x 0 = decompose 40 into 4 x 10, decompose 0 into x 10 create an equation using the decomposed numbers 4 x 10 x x 10 = group ones and tens (4 x ) x (10 x 10) 8 x 100 = 800 OBJECTIVE OF TOPIC B Focus Area Topic B Multiplication by 10, 100, and 1,000 Place Value Chart & Number Disks Use number disks to represent 143 First, draw 1 circle in the hundreds place to show 1 hundreds. Next draw 4 circles in the tens place to show 4 tens. Finally, draw 3 circles in the ones place to show 3 ones. Use a place value chart to multiply Start by creating number disks to represent 1 one. (the black circle). Place a circle around the group of 1 ones to show that the group will moving as a whole. To show that 1 one is being multiplied by ten, draw an arrow to the tens place, and re-draw the group of 1. Because it was multiplied by 10 it is no longer 1 one, it is now 1 ten. The circles are drawn differently in order to show which number disks have been moved already. Another way to look at it is having 1 group of 10 ones. 10 ones is equal to 1 ten. On this chart bundle the 10 ones to make 1 ten. 10 x 1 = 10. The same concept applies when multiplying 15 x 10. Draw 15 on the place value chart. 1 ten and 5 ones Multiply 5 ones by ten to get 5 tens. (ones x tens = tens) Multiply 1 ten by tens to get 1 hundred. (tens x tens = hundreds) 15 x 10 = 1 hundred 5 tens 0 ones or 10 x 15 = 150 Area Model 1 3 Interpret and represent patterns when multiplying by 10, 100, and 1,000 in arrays and numerically. Multiply multiples of 10, 100, and 1,000 by single digits, recognizing patterns. Multiply two-digit multiples of 10 by two-digit multiples of 1- with an area model.

4 Grade 4, Module 3, Topic C Math ( 013 Common Core, Inc.) that is also posted as the newsletter will discuss Module 3, Topic C. Topic C. Multiplicative of up to Four Digits by Single- Digit Numbers Words to know OBJECTIVE OF TOPIC C Partial Products Standard Algorithm Things to Remember!!! o o o To regroup or bundle a group of 10 ones means to represent it as 1 ten. To regroup or bundle a group of 10 tens means to represent it as 1 hundred Commutative Property is when numbers can be swapped but the answer is the same. Use place value disks to represent two-digit by one-digit multiplication. Extend the use of place value disks to represent three- and four-digit by one-digit multiplication. Multiply three- and four-digit numbers by one-digit numbers applying standard algorithm. Connect area model and partial products method to standard algorithm. Focus Area Topic C Multiplicative of up to Four Digits by Single-Digit Numbers Represent x 14 with disks Begin by drawing disks to represent 14. Look at the number of times 14 is multiplied by,. So repeat the pattern twice. The chart should have 14 represented twice on the place value chart. Now add the ones together. 4 ones + 4 ones = 8 ones. Next add the tens together. 1 ten + 1 ten = tens. tens + 8 ones = = 8. Represent x 14 with partial products A partial product is written vertically. 14 x =? First multiply the ones column. 4 ones x ones = 8 ones. Next multiply the tens column. 1 ten x tens = tens. 8 ones + tens = 8 Represent 5 x 4 with disks Begin by drawing disks to represent 4. Look at the number of times 4 is multiplied by, 5. So repeat the pattern five times. The chart should have 4 represented five times on the place value chart. Look in the ones place to see if bundling could be used. Yes, there are groups of 10 ones that can be changed to tens. Circle the groups, place an arrow showing that those groups of 10 ones will be moved to the tens place, then draw the circles to represent tens in the tens place. Look in the tens place to see if bundling could be used. Yes, there is 1 group of 10 tens that can be changed to 1 hundred. Circle the group, place an arrow showing that the group of 10 tens will be moved to the hundreds place, then draw the circles to represent 1 hundred in the hundreds place. Now add the ones together. There are 0 ones in the ones place. Add the tens together. There are 4 tens in the tens place. Add the hundreds together, there is 1 hundred in the hundreds place. 1 hundred + 4 tens + 0 ones = 140.

5 Solve and represent 3 x 951 using a partial products drawing on the place value chart. Record the partial product when multiplying each unit. 1 one x 3 ones = 3 ones, draw 3 disks in the ones place. 5 tens x 3 tens = 15 tens = 1 hundred + 5 tens, draw one disk in the hundreds place and 5 disks in the tens place. 9 hundreds x 3 hundreds is 7 hundreds = thousands + 7 hundreds, draw disks in the thousands place and 7 disks in the hundreds place. Add up the disks in each column of the place value chart then write the total number of each column under the chart in the appropriate column. Solve and represent 3 x 56 in a place value chart and relate the process to solving using the standard algorithm. First draw the number disks to represent 56. It is multiplied by 3 so draw two more sets of 56 to show that the number is multiplied by 3. Go though the sets to bundle 10 s as needed. Write the number at the bottom of the place value chart in the appropriate place. Let s look at the place value chart and compare it to the standard algorithm. In the ones column there were 18 ones. We regrouped (bundled) 10 ones for 1 ten that left 8 ones. That is similar to what was done in the standard algorithm. 6 x 3 = 18, put an 8 in the ones place and put the 1 ten on top of the 5 in the tens place. This same concept occurred in the tens place. 5 x = 16. Write 6 tens in the tens column and 1 hundred was carried to the hundreds column. This process continues until there are no more numbers to multiply. NOTE: Both ways to solve the standard algorithm is correct. Solve 5 x 358 using a partial product algorithm and the standard algorithm and relate the two methods. In partial product algorithm, when multiplying the ones, the ones are written on the first line, when multiplying the tens, the tens are written on the second line, and so on. When using the standard algorithm, in the ones column only the ones are written, the tens are written on top or beneath the tens column. For example, in the standard algorithm, when multiplying 8 x 5, there are 4 tens and 0 ones. Write the 0 under the ones column and write the 4 tens on top of the tens column. Now multiply 5 x 5. 5 x 5 = 5, but there are still 4 tens to add to the 5 tens. So hundreds + 5 tens + 4 tens = 9 tens = hundreds and 9 tens. Write the 9 in the tens column and write the in the hundreds column. And continue the process until there are no more numbers to multiply and add. Writing the numbers on top or on the bottom of the problem is correct as long as it is in the correct column. Word problems using a tape diagram and standard algorithms Jonas and Cindy are making candied apples. Jonas purchased 534 grams of apples. Cindy purchased 3 times as many grams of apples. How many grams of apples did they purchase together?

6 Grade 4, Module 3, Topic D Math ( 013 Common Core, Inc.) that is also posted as the newsletter will discuss Module 3, Topic D. Topic D. Multiplication Word Problems Things to Remember!!! Focus Area Topic D Multiplication Word Problems Multi-Step Problems The table shows the cost of party favors found in 1 party bag. Each guest receives balloons, 3 lollipops, and 1 bracelet. What is the total cost for 8 guests? One bag = $1.18 Item Cost 1 balloon 4 1 lollipop 1 1 bracelet 34 o Read the word problem carefully to figure out what steps are needed to solve each problem. balloons 4 x = 48 3 lollipops 1 x 3 = 36 1 bracelet 34 x 1 = 34 They paid for the party favors with a $0 bill. How much change should they expect back? It takes 5 more to get to 150 and 600 more to get to 750. OBJECTIVE OF TOPIC D 1 Solve two-step word problems, including multiplicative comparison. Use multiplication, addition, and subtraction to solve multi-step word problems. They would receive $10.56 change.

7 Grade 4, Module 3, Topic E Math ( 013 Common Core, Inc.) that is also posted as the newsletter will discuss Module 3, Topic E. Topic E. Division of Tens and Ones with Successive Remainders Words to know dividend number bond divisor area model quotient standard division remainder tape diagram array place value chart Focus Area Topic E Division of Tens and Ones with Successive Remainders Modeling a Division Problem There are 15 students in Science class separated into 5 groups. How many students are in each group? Model with an Array How many students all together? 15 How many groups of students? 5 How many per group? 3 Start by creating the 5 groups, draw 5 larger circles. Next ask yourself Do I have enough to give every group one student? Yes, you can place one student in each group. Continue until there are no more students to group. 15 has no remainder when divided into 5 groups. To check your work skip count by 5 s to 15. 5, 10, 15. Or 5 x 3 = 15 Things to remember!!! Always label your work when creating an area model. OBJECTIVE OF TOPIC E The remainder represents the amount left over after dividing. For example 16 cannot be divided exactly by 5. The closest you can get without going over is 5 x 3 =15 which is 1 less than = 3 r1 1 Solve division word problems with remainders. 3 Understand and solve division problems with a remainder using the array and area model. Understand and solve two-digit dividend division problems with a remainder in the ones place by using number disks. 4 Represent and solve division problems requiring decomposing a remainder in the tens. Place Value Disks are circles with a number written inside of them in order to represent place value. represents ones place, represents tens place and represents hundreds place. 5 Find whole number quotients and remainders. 6 7 Explain remainders by using place value and understanding and models. Solve division problems without remainders using the area model. 8 Solve division problems with remainders using the area model.

8 Modeling a Division Problem with Remainders In Lesson 14 students will represent a division problem using an array, a number bond, and a tape diagram. There are 16 students in a Science class separated into 5 groups. How many students are in each group? Three important questions How many students all together? How many groups of students? How many per group? Represent using an array Start by creating the 5 groups, draw 5 larger circles. Next ask yourself Do I have enough to give every group one student? Yes, you can place one student in each group. There were 16, 1 student was placed in each group, so 16 5 = 11. There are 11 students that still need to be placed into groups. Continue this process until each student is placed. The 1 left is the remainder. Represent using a number bond The top circle is always the total number, in this case 16. The number on the left is always groups that can be made. This number will always be the highest multiple of the group. 15 is the largest multiple of 5, which does not go over the total number of students. The number on the right is always the remainder. The amount left over after the number is divided evenly. Represent using a tape diagram The tape diagram is similar to the array, instead of circles there are numerals. In this tape diagram the bar is separated into 5 sets of 3 s. Skip count by three 5 times. 3, 6, 9, 1, plus the remainder of = 16. That is the number of students the Science class. This is a way that the answer can be checked. Another way is to multiply. 5 x 3 = 15 Next add the remainder = 16. These are great ways to check your work. Modeling a Division Problem with Remainders In Lesson 15 students will represent a division problem using an area model. Represent using an area model An area model is faster to draw and it represents the same division problem. The total number is 16. That is the area of the model. The number of groups is 5. That is the length or the width of the area model. Mark off squares 5 at a time, count 5, 10, 15. Now we have to represent the remainder. Draw one more box to represent the remainder of 1. The total number of squares is 16. The quotient is 3 the remainder is 1. In Lesson 16 students will represent a division problem using standard division and a place value chart. Represent using standard division Standard division is just dividing using numerals. What number can be multiplied by 5 and is the closest to 16? 3. 5 x 3 = 15, write that number below the 16. Subtract = 1. 5 cannot be divided into 1 so this is your remainder. Represent using place value chart Divide the bottom of the place value chart into the number of groups needed. For this problem it is divided into 5 groups. Start with the largest place value group, you have 1 ten. Can I separate 1 ten into 5 groups? No, so we decompose the 1 ten into 10 ones. Now fair share the 16 ones into the 5 groups. Remember to mark off each one as you place it in a group. There are 3 ones in each group and 1 remaining which has not been placed in a group. The answer is 3 r1 or 3 remainder 1.

9 Grade 4, Module 3, Topic F Math ( 013 Common Core, Inc.) that is also posted as the newsletter will discuss Module 3, Topic F. Topic F. Reasoning with Divisibility Words to know Factor Composite Number Products Prime Number Multiple Associative Property Focus Area Topic F Reasoning with Divisibility Identify Factors and Product What are the two multiplication sentences that represent the arrays above? 1 x 6 = 6 and x 3 = 6 The same product is represented in both sentences. What are the factors of 6? 1,, 3, 6 Look at the list of factors, draw an arrow to connect the factor pairs. Things to remember!!! The Commutative Property says you can swap numbers (or change order) and still get the same answer. 1 x 6 = 6 and 6 x 1 = 6 Notice that and 3 are the middle factor pair. We have checked all numbers up to. There are no numbers between and 3, so we have found all factors of 6. 1 x 5 = 5 Find another factor pair for 5. 5 x 1 = 5 OBJECTIVE OF TOPIC F 1 Find factor pairs for numbers to 100 and use understanding of factors to define prime and composite. Use division and the associative property to test for factors and observe patterns., 3, and 4 are not factors of 5, so 5 has only one set of factors. Numbers that have exactly two factors, 1 and itself are called prime numbers. Numbers that have at least one other factor beside 1 and itself are called composite numbers. Factors can also be written in a table. 3 4 Determine whether a whole number is a multiple of another number. Explore properties of prime and composite numbers to 100 using multiples

10 Use division to find factors of larger numbers. How can one find out if 3 is a factor of 48? Divide 48 by 3. What if there is a remainder? If there is a remainder then 3 is not a factor of is a factor of 48 because there are no remainders when divided. Use the associative property to find additional factors Find the factors of 48. Is 5 a factor of 48? No, any number multiplied by 5 ends with a 0 or a 5. Is a factor of 48? Yes, is a factor of all even numbers. Is 1 a factor of 48? Yes, 1 is a factor of all numbers. Is 3 a factor of 48? Yes, we divided 48 by 3 and had no remainders. Is 6 a factor of 48? Yes, 6 x 8 = 48 Is this number sentence true? 48 = 6 x 8 = ( x 3) x 8 What is a multiple? Count by 3 s to 30. 0, 3, 6, 9, 1, 15, 18, 1, 4, 7, 30 What pattern is being used when counting? Add 3 to the number said When we skip count by a whole-number, the numbers said are called multiples. How are multiples different from factors. When listing factors, we listed them and were done, multiples can go on forever. Is 84 a multiple of 1? Yes, 1 x 7 = 84 or count 1, 4, 36, 48, 60, 7, 84 Using the associative property, since 3 x 4 = 1 we also know that 84 is also a multiple of 3 and 4. We also know that 3, 4, 8, and 1 are also factors of x 6 = 4 x ( x 3) is the original problem The associative property says that when we are multiplying all numbers together we can multiply the numbers in any order and still get the same answer. In the problem above, we can move our parentheses and multiply 4 x first then multiply the answer by 3. 4 x = 8 and 8 x 3 = 4. Use the associative property to see that and 3 are both factors of 48. The associative property means that it does not matter how you group numbers when you multiply. x 3 = 6 Move the parentheses so that 3 is associated with the 8 instead of the. 3 x 8 = 4 and 4 x = 48 3 x = 6 Move the parentheses so that is associated with the 8 instead of the 3. x 8 = 16 and 16 x 3 = 48 The commutative property states that you can swap numbers over or change the order of the numbers and the answer will remain the same, so x 3 = 6 and 3 x = 6. We know that we can use the associative property next to solve the problem.

11 Grade 4, Module 3, Topic G Focus Area Topic G Division of Thousands, Hundreds, Tens & Ones Math ( 013 Common Core, Inc.) that is also posted as the newsletter will discuss Module 3, Topic G. Topic G. Division of Thousands, Hundreds, Tens, & Ones Place Value Charts and Words to know place value chart number bond standard division area model tape diagram decompose Draw 6 ones, divide it into 3 groups. There are ones in each group. Draw 6 tens, divide it into 3 groups. There are tens in each group. Regrouping with a place value chart OBJECTIVE OF TOPIC G Divide multiples of 10, 100, and 1,000 by singledigit numbers. Represent and solve division problems with up to a three-digit dividend numerically and with number disks requiring decomposing a remainder in the hundreds place. Represent and solve three-digit dividend division with divisors of, 3, 4, and 5. Represent numerically four-digit dividend division with divisors of, 3, 4, and 5, decomposing a remainder up to three times. Solve division problems with a zero in the dividend or with a zero in the quotient. Interpret division word problems as wither number of groups unknown or group size unknown. Interpret and find whole number quotients and remainders to solve one-step division word problems with larger divisors of 6, 7, 8, and 9. Explain the connection of the area model of division to the long division algorithm for threeand four-digit dividends. Notice on the place value chart in the top row on the top line the value is 54. When dividing being with the tens place on the place value chart. 50 is divided into 4 groups (each row represents one group). Place 1 ten in each group. This leaves 1 ten that cannot be divided evenly into 4 groups. Circle the ten and decompose it to 10 ones, making sure to circle the ten and draw the arrow to show that it has been moved to the ones place. Next divide the 14 ones into 4 groups. Notice the line drawn through the circles on the top row, this is to help students remember if the number (circle) has been counted already when dividing. Each group has 3 ones and there are ones remaining. 4 can be divided into 54 how many times? 1 ten and 3 ones remainder ones or 13 r times

12 The Jonesville Hotel has a total of 600 rooms. That is 3 times as many rooms as the Donaldsville Hotel. How many rooms are there in the Donaldsville Hotel? Draw a tape diagram to model this problem. In one day, the donut shop made 719 chocolate donuts. They sold all of them by the dozen. A few donuts were left over and the baker took them home. How many donuts did the baker take home? The Thomasville High School is replacing the seats in the football stadium. They purchased 750seats and 34 seats were donated. There are 3 sections for seats and they want to place the same number of seats in each section. How many seats would be in each section? How many seats do they have left? First find the total number of seats = 784 Next divide to solve the problem. Each section will have 61 seats and there will be one seat left that will not be used in the stadium. Look at the image above, a tape diagram is drawn. A tape diagram uses a rectangle(s) with numbers to represent the number in a word problem. Now that numbers are getting bigger a rectangle is used to represent the number instead of drawing dots or pictures. A tape diagram allows the student to visualize the problem. The image also has a sample of a standard division problem and a place value chart. Students can use various tools to solve word problems. Students will compare standard division to a tape diagram and find the relations between the two tools used for solving division problems. There are 719 donuts sold in sets of 1. The donut shop sold 59 boxes of donuts and the baker took 11 donuts home. Students will also learn how to divide using number bonds and area models. Drawing an area model to solve : Draw a rectangle with a width of 6 (This is the known side). Six times how many hundreds gets us as close as possible to an area of 100? hundreds. (00 x 6 = 100) How many hundreds remain? Zero. ( = 4) We have 4 units left with a width of 6. Six times how many units gets us close to 4 tens? 5 ones. (5 x 6 = 30) Add 5 ones to the length. How many tens remain? 1 ten ones. (4 30 = 1) We have 1 units remaining. Six times how many units gets us close to 1 ten ones? ones. ( x 6 = 1) How many remain? Zero. Then length of the unknown side is = 07 Create a number bond to solve : A number bond is similar to an area model. Follow the same steps as the area model. How many hundreds, tens, etc? Recording the numbers as the problem is being solved. Look at the number bond that is separated into 5 bonds. Sometimes it is easier to divide with smaller numbers. Not all students will decompose the numbers in the same way, but as long as the number bonds add up to the number they are decomposing the answer will remain the same = 14 and = 14. When dividing both answers will be 07.

13 Grade 4, Module 3, Topic H Math ( 013 Common Core, Inc.) that is also posted as the newsletter will discuss Module 3, Topic H. Focus Area Topic H Multiplication of Two-Digit by Two-Digit Numbers Multiply using an area model and partial product Topic H. Multiplication of Two-Digit by Two-Digit Numbers Students are introduced to the multiplication algorithm for two-digit by two-digit numbers. The lessons in Topic H provide a firm foundation for understanding the process of the algorithm. Students will make a connection from the area model to the partial product to the standard algorithm. Multiply using an area model standard algorithm 54 x 4 Multiply using a place value chart Draw disks to show. Draw arrows to show 10 times that amount. Draw 4 groups of to represent 4 times that amount. Solve: 8 hundreds 8 tens 4 x (10 x ) = 880 OBJECTIVE OF TOPIC H Multiply two-digit multiples of 10 by two-digit numbers using a place value chart. Multiply two-digit multiples of 10 by two-digit numbers using the area model. Multiply two-digit by two-digit numbers using four partial products. Transition from four partial products to the standard algorithm for two-digit by two-digit multiplication. Draw a rectangle. Write the numbers in expanded form, or in each place value. This will determine how to subdivide the rectangle vertically and horizontally Label the area model. Write the expressions that represent the area in each of the smaller rectangles and solve each of those equations. Add the product of the first row together. ( x 4) + ( x 50) = = 108 Add the products of the second row together. (40 x 4) + (40 x 50) = =,160 Next add the sum of both rows together ,160 =,68 54 x 4 =,68