Schoolmaster Miller's Classroom - Home



5600700-342900Eureka Math: Mod. 1 L. 1MATH CCSS: 5.NBT.A.10Eureka Math: Mod. 1 L. 1MATH CCSS: 5.NBT.A.1685800-114300COMPARE DECIMAL PLACE VALUE0COMPARE DECIMAL PLACE VALUE 68580067119500-11430099695Standard: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.0Standard: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.457200129540The standard wants you to recognize that the 6 in the hundreds place is 10 times bigger than the 4 in the tens place, and that the 6 in the hundreds place is 1/10 of what the 3 represents in the thousands place.00The standard wants you to recognize that the 6 in the hundreds place is 10 times bigger than the 4 in the tens place, and that the 6 in the hundreds place is 1/10 of what the 3 represents in the thousands place.0214630EXAMPLE00EXAMPLE-800100379095In which number does the digit 8 have a value that is 1/10 times as great as the digit 8 in the number 2,980.7?0In which number does the digit 8 have a value that is 1/10 times as great as the digit 8 in the number 2,980.7?2514600799465The value of the digit 8 in 2,980.7 is in the tens place, so 1/10 times as great means I am decreasing one place value which would be the ones place, so the 8 in 548 is 1/10 times as great as the 8 in 2,980.7. In other words 8 is 1/10 times as great as 80.00The value of the digit 8 in 2,980.7 is in the tens place, so 1/10 times as great means I am decreasing one place value which would be the ones place, so the 8 in 548 is 1/10 times as great as the 8 in 2,980.7. In other words 8 is 1/10 times as great as 80.800100799465a) 3,023.8b) 8,659c) 389.3d) 5480a) 3,023.8b) 8,659c) 389.3d) 5482171700808355The value of the digit 5 in 4.59 is 5 tenths, so a number that is ten times greater means the 5 has to be in the one’s place. 5 is 10 times as great as 0.5.00The value of the digit 5 in 4.59 is 5 tenths, so a number that is ten times greater means the 5 has to be in the one’s place. 5 is 10 times as great as 0.5.228600236855In which number does the digit 5 have a value that is 10 times as great as the digit 5 in the number 4.59?0In which number does the digit 5 have a value that is 10 times as great as the digit 5 in the number 4.59?914400744220a) 578b) 31.757c) 2,775.9d) 134.5910a) 578b) 31.757c) 2,775.9d) 134.5912171700137160ERROR ANALYSISa) the 5 is in the hundreds place, b) the 5 is in the tenths place, c) the 5 is in the ones place, and d) the 5 is in the tenths place. 00ERROR ANALYSISa) the 5 is in the hundreds place, b) the 5 is in the tenths place, c) the 5 is in the ones place, and d) the 5 is in the tenths place. 45720043180To master this standard, you really need to be familiar with the place value chart, and be able to recreate one at any given point.00To master this standard, you really need to be familiar with the place value chart, and be able to recreate one at any given point.605790014605BASE TEN0BASE TEN5600700-342900Eureka Math: Mod. 1 L. 1MATH CCSS: 5.NBT.A.10Eureka Math: Mod. 1 L. 1MATH CCSS: 5.NBT.A.1457200-114300VALUE OF A DIGIT0VALUE OF A DIGITStudents need to be able to distinguish the value of a digit within its position in the place value chart. COMMON MISUNDERSTANDINGS AMONG 5TH GRADERSPlace value can be pretty abstract especially when we are discussing very large numbers in the millions, and especially anything smaller than hundredths. For example consider this common comparison: Which is greater? .2 or .002Hopefully most students will understand that .2 is a greater number than .002, but I think that lots of students hear themselves say tenths and thousandths and quickly assume that thousandths is larger, when in reality it is much smaller. Students need to understand that .002 is 100 times smaller than .2.422910040640Compare and order the list of numbers from least to greatest:.039, .39, 3.9, .390 .039 .39 3.9 .390 .039 .39 3.9 .3900.0390.3903.9000.390The numbers from least to greatest are: .039, .39, .390, 3.90Compare and order the list of numbers from least to greatest:.039, .39, 3.9, .390 .039 .39 3.9 .390 .039 .39 3.9 .3900.0390.3903.9000.390The numbers from least to greatest are: .039, .39, .390, 3.9First of all, students need to be familiar with a place value chart, and should be able to recreate one at any opportunity to do so.5029200408940First, I need to preserve place value, so I’ll line up the decimal points.00First, I need to preserve place value, so I’ll line up the decimal points.Secondly, when comparing two or more numbers, students should stack them in such a way as to preserve the place value. Next step is to fill in empty spaces with zero.-11430024130EXAMPLE00EXAMPLE537210038735Here I can see that I have 3 wholes, so that’s my greatest number. Now I am going to fill in the missing place values with zero.00Here I can see that I have 3 wholes, so that’s my greatest number. Now I am going to fill in the missing place values with zero.The digit 2 in which number represents a 1371600122555ERROR ANALYSISThe digit 2 is in the thousandths place in the question, and in a) the 2 is also in the thousandths place. In b) the 2 is in the ones place, c) the 2 is in the tenths place, and in d) the 2 is in the ones and tenths place.00ERROR ANALYSISThe digit 2 is in the thousandths place in the question, and in a) the 2 is also in the thousandths place. In b) the 2 is in the ones place, c) the 2 is in the tenths place, and in d) the 2 is in the ones and tenths place.value of 0.002?228600122555a) 8.032b) 162c) 0.324d) 72.2300a) 8.032b) 162c) 0.324d) 72.2305029200122555Now working from left to right, I can clearly see that 3.9 is my greatest number, because the other whole numbers in place value are 0.00Now working from left to right, I can clearly see that 3.9 is my greatest number, because the other whole numbers in place value are 0.434340079375The second and fourth numbers are equal at 0.390, which is easy to see once the zero is added. This leaves the number 0.039 as the least because obviously the zero in the tenths place is less than 3.00The second and fourth numbers are equal at 0.390, which is easy to see once the zero is added. This leaves the number 0.039 as the least because obviously the zero in the tenths place is less than 3.What value does the 4 represent in thenumber 487.009?137160022860ERROR ANALYSISThe value of 4 in the question is 400 hundred, so the only answer is b) 400. a) is the wrong value, c) is a misread of the decimal point and comma, and d) demonstrates a misunderstanding of place value.00ERROR ANALYSISThe value of 4 in the question is 400 hundred, so the only answer is b) 400. a) is the wrong value, c) is a misread of the decimal point and comma, and d) demonstrates a misunderstanding of place value.228600113030a) 40b) 400c) 400,000d) .4000a) 40b) 400c) 400,000d) .4006057900108585BASE TEN0BASE TEN-114300-114300MULTIPLY AND DIVIDE BY POWERS OF 1000MULTIPLY AND DIVIDE BY POWERS OF 105486400-342900Eureka Math: Mod. 1 L. 2MATH CCSS: 5.NBT.A.20Eureka Math: Mod. 1 L. 2MATH CCSS: 5.NBT.A.2Standard: Students should be able to explain patterns when multiplying or dividing by powers of 10 especially when it involves the movement of the decimal point. Students should understand that every number has a decimal point whether it is seen or not, and that decimal point moves to the right for multiplication and to the left for division.Students should be able to explain why they are moving the decimal point one way or another and not just memorizing a little math trick.502920069850In this lesson, students used the place value chart and based on multiplying or dividing by 10, 100, or 1000 use arrows to designate a digits new position in place value.0In this lesson, students used the place value chart and based on multiplying or dividing by 10, 100, or 1000 use arrows to designate a digits new position in place value.306615193807003180476586966I think it’sactually easier to add a 0 or take a 0 depending on the operation.0I think it’sactually easier to add a 0 or take a 0 depending on the operation.22860001148715EXAMPLE00EXAMPLE-228600691515002286008058150011430069151500 Solve.a) 36,000 ? 10240030064135In this case, 36,000 is placed correctly in the place value chart and because we are multiplying by 10, we are moving one place value to the left to make it 360,000. If we multiplied by 100 we’d move two spaces to the left, and our answer: 3,600,00000In this case, 36,000 is placed correctly in the place value chart and because we are multiplying by 10, we are moving one place value to the left to make it 360,000. If we multiplied by 100 we’d move two spaces to the left, and our answer: 3,600,0002400300184785Likewise, because this is division we’d move one space to the right and our answer would be 3,600. And if we divided by 100 our answer would be 360.00Likewise, because this is division we’d move one space to the right and our answer would be 3,600. And if we divided by 100 our answer would be 360.b) 36,000 ÷ 10605790029845BASE TEN0BASE TEN5486400-342900Eureka Math: Mod. 1 L. 3MATH CCSS: 5.NBT.A.20Eureka Math: Mod. 1 L. 3MATH CCSS: 5.NBT.A.2228600-114300POWERS OF 10 EXPLAINEDby Mathematics-00POWERS OF 10 EXPLAINEDby Mathematics-68580014986000331470085725The small 2 written beside the 10 means it is raised to an exponent of 2. This means 10 is multiplied by itself 2 times.The small 2 written beside the 10 means it is raised to an exponent of 2. This means 10 is multiplied by itself 2 times.3657600106680102 = 10 ? 10 = 100103 = 10 ? 10 ? 10 = 1,000104 = 10 ? 10 ? 10 ?10 = 10,000105 = 10 ? 10 ?10 ? 10 ? 10 = 100,0000102 = 10 ? 10 = 100103 = 10 ? 10 ? 10 = 1,000104 = 10 ? 10 ? 10 ?10 = 10,000105 = 10 ? 10 ?10 ? 10 ? 10 = 100,000228600-127000377190034290Notice how 10 to the power of 3 is 1,000. There are 3 zeroes after the 1. Check out that pattern.0Notice how 10 to the power of 3 is 1,000. There are 3 zeroes after the 1. Check out that pattern.3771900120015If we understand that all numbers have a decimal point whether it is seen or not, then we can also divide by powers of 10.0If we understand that all numbers have a decimal point whether it is seen or not, then we can also divide by powers of 10.3657600140970EXAMPLE00EXAMPLE22860012636500Write the following in standard form(e.g. 5 ? 103 = 5,000)MULTIPLICATION 72 ? 104 = 72 ? 10,000 = 720,000 4.036 ? 102 = 4.036 ? 100 = 403.6DIVISION 7,600 ÷ 102 = 7,600 ÷ 100 = 76 2,800,000 ÷ 104 = 2,800,000 = 280 10,00022860052705SCIENTIFIC NOTATIONIn science we deal with very large numbers. The first factor is a number less than 10 and the second factor is a power of ten. The distance from earth to the sun is 150,000,000 kilometers. If we were to express that distance in scientific notation it would look like 1.5 ? 108.00SCIENTIFIC NOTATIONIn science we deal with very large numbers. The first factor is a number less than 10 and the second factor is a power of ten. The distance from earth to the sun is 150,000,000 kilometers. If we were to express that distance in scientific notation it would look like 1.5 ? 108.605790091440BASE TEN0BASE TEN5600700-457200Eureka Math: Mod. 1 L. 4MATH CCSS: 5.NBT.3MATH CCSS: 5.MD.A.10Eureka Math: Mod. 1 L. 4MATH CCSS: 5.NBT.3MATH CCSS: 5.MD.A.1457200-228600USING THE METRIC SYSTEMWITH POWERS OF TEN00USING THE METRIC SYSTEMWITH POWERS OF TENStandard: Students will apply what they have learned in multiplying and dividing by “Powers of 10” to convert units using the metric system. Students should understand that in 1 meter there are 10 decimeters, or 100 centimeters, or 1000 millimeters. The same is true for measuring volume in the metric system, which utilizes the root of liter. 4114800352425MNEMONIC DEVICE“King Hector died Monday drinking chocolate milk,” is mnemonic device.00MNEMONIC DEVICE“King Hector died Monday drinking chocolate milk,” is mnemonic device.Students will apply the “Powers of 10,” movement of the decimal point, when working within the Metric System.44405552513330Mr. Miller calls this,“Level the playing field.”00Mr. Miller calls this,“Level the playing field.”40870382023977004114800790575Using the “Powers of 10,” students should be able to find that 350 ml is .35L by dividing 350 ÷ 103 = .350.00Using the “Powers of 10,” students should be able to find that 350 ml is .35L by dividing 350 ÷ 103 = .350.03175EXAMPLE00EXAMPLEArrange the following measurements in order from smallest to largest.3657600892175STEP 4:I could also chop off the zeroes for both the dividend and divisor. 32,000 ÷ 1000 = 3200STEP 4:I could also chop off the zeroes for both the dividend and divisor. 32,000 ÷ 1000 = 321828800892175SOLVEI could set up my work as 32,000 ÷ 1000 and do the traditional algorithm. I could set it up and solve as a “Power of 10,” 32,000 ÷ 103 = 32, and move the decimal 3 places to the left.00SOLVEI could set up my work as 32,000 ÷ 1000 and do the traditional algorithm. I could set it up and solve as a “Power of 10,” 32,000 ÷ 103 = 32, and move the decimal 3 places to the left.228600892175UNDERSTANDING32,000 is a whole number, so it has a decimal point after the last zero. There are 1000 mL in one liter, so I know that I need to divide 32,000 by 1000.00UNDERSTANDING32,000 is a whole number, so it has a decimal point after the last zero. There are 1000 mL in one liter, so I know that I need to divide 32,000 by 1000.228600320675Remember that when we compare and order numbers we want to stack them, but in the above problem we must first turn 32,000 mL to liters so that we can truly compare.LITERS03.032.002.10Remember that when we compare and order numbers we want to stack them, but in the above problem we must first turn 32,000 mL to liters so that we can truly compare.LITERS03.032.002.13 liters32,000 milliliters2.1 liters2286002202815It is important to “Level the playing field.”0It is important to “Level the playing field.”36576001745615Once stacked, it’s easy to see 32L is largest and then obviously 2.1L is smaller than 3L.In order from smallest to largest: 2.1L, 3L, 32L00Once stacked, it’s easy to see 32L is largest and then obviously 2.1L is smaller than 3L.In order from smallest to largest: 2.1L, 3L, 32L605790037465BASE TEN0BASE TEN228600-114300DECIMALS IN WORD (WRITTEN) FORM00DECIMALS IN WORD (WRITTEN) FORM5600700-342900Eureka Math: Mod. 1 L. 5MATH CCSS: 5.NBT.A.3MATH CCSS: 5.NBT.A.3.A0Eureka Math: Mod. 1 L. 5MATH CCSS: 5.NBT.A.3MATH CCSS: 5.NBT.A.3.A4229100149860TO “AND” OR NOT TO “AND”Sometimes kids will add the word “and” when saying a number like One thousand three hundred and seventy two, but when saying and writing numbers we want to reserve the word “and” to denote a decimal point. If we use “and” incorrectly, we run the risk of being inaccurate. That number looks like this: 1,300.72But without the word hundredths following it just sounds wrong.Let’s say we are writing a check for rent using the above number. Written correctly in word form it would look like this: One thousand three hundred seventy-two and no cents (meaning nothing after the decimal point).0TO “AND” OR NOT TO “AND”Sometimes kids will add the word “and” when saying a number like One thousand three hundred and seventy two, but when saying and writing numbers we want to reserve the word “and” to denote a decimal point. If we use “and” incorrectly, we run the risk of being inaccurate. That number looks like this: 1,300.72But without the word hundredths following it just sounds wrong.Let’s say we are writing a check for rent using the above number. Written correctly in word form it would look like this: One thousand three hundred seventy-two and no cents (meaning nothing after the decimal point).Standard: Students need to be able to read, write, and compare decimals to thousandths.Students need to be able to read and write numbers up to three place values behind the decimal point.Two-Step Process1) Say the entire number behind the decimal point. For instance: .007 can be said as seven .092 can be said as ninety-two .248 can be said as two hundred forty-eight2) Say the place value of the last digit. In the examples above they all end in the thousandths column on a place value chart. So the number said correctly is: .007 is said as seven thousandths .092 is said as ninety-two thousandths .248 is said as two hundred forty-eight thousandths91440093980DECIMALS IN EXPANDED FORM00DECIMALS IN EXPANDED FORMWe learned that expanded form for 347 was 300 + 40 + 7, and maybe we learned it as 3 ? 100 + 4 ? 10 + 7 ? 1 = 347. Both are examples of expanded form. This lesson is no different.Students will be able to use fractions or decimals to express the decimal place value units. Hundred ThousandsTen ThousandsThousandsHundredsTensOnes?TenthsHundredthsThousandths100,00010,0001,000100101?.1.01.001100,00010,0001,000100101?110110011000??34290048895EXAMPLE00EXAMPLEWrite an expression that represents the number 2,478.602In fraction form: (2 ? 1000) + (4 ? 100) + (7 ? 10) + (8 ? 1) + (6 ? 1/10) + (2 ? 1/1000)605790019050BASE TEN0BASE TENIn decimal form: (2 ? 1000) + (4 ? 100) + (7 ? 10) + (8 ? 1) + (6 ? .1) + (2 ? .001)342900-114300COMPARE DECIMALS TO THOUSANDTHS00COMPARE DECIMALS TO THOUSANDTHS5600700-342900Eureka Math: Mod. 1 L. 6MATH CCSS: 5.NBT.A.3.B0Eureka Math: Mod. 1 L. 6MATH CCSS: 5.NBT.A.3.BStandard: Students will compare and possibly order two decimals to thousandths using <, =, or > symbols to record their comparisons.Anytime we are comparing two numbers, we should do three things: 1) We should stack them by preserving place value with the decimal point 2) We should fill in any gaps with zero 3) Then compare each digit from left to rightThe symbols we use for greater than or less than look like this: < and >, so which is which? Use the hint: “The alligator always goes for the greater meal.”02240280EXAMPLE00EXAMPLE2971800411480or0or Use <, >, or = to compare the following. 16.45 and 16.4540577851) Stack them: 2) Fill gaps in with zeroes 3) Compare left to right 16.45 16.450 1 6 . 4 5 0 16.454 16.454 1 6 . 4 5 4The last digits circled are different, so 16.45 < 16.454001) Stack them: 2) Fill gaps in with zeroes 3) Compare left to right 16.45 16.450 1 6 . 4 5 0 16.454 16.454 1 6 . 4 5 4The last digits circled are different, so 16.45 < 16.45456089363864600Use <, >, or = to compare the following. 419.10 and 419.099Some students who don’t complete the steps may say that .10 is less than .099, so its super important to take the time in a real comparison by following the steps above.0736601) Stack them: 2) Fill gaps in with zeroes 3) Compare left to right 419.10 419.10 4 1 9 . 1 0 0 419.099 419.099 4 1 9 . 0 9 9The digits in the tenths place are circled because the difference in numbers is there, and obviously there is a difference between 1 and 0, so 419.10 > 419.099001) Stack them: 2) Fill gaps in with zeroes 3) Compare left to right 419.10 419.10 4 1 9 . 1 0 0 419.099 419.099 4 1 9 . 0 9 9The digits in the tenths place are circled because the difference in numbers is there, and obviously there is a difference between 1 and 0, so 419.10 > 419.099549448657858005943600134620BASE TEN0BASE TEN342900-114300ROUNDING: WHOLE AND DECIMALS00ROUNDING: WHOLE AND DECIMALS5600700-342900Eureka Math: Mod. 1 L. 7-8MATH CCSS: 5.NBT.A.40Eureka Math: Mod. 1 L. 7-8MATH CCSS: 5.NBT.A.4Standard: Students should be able to use place value to round whole numbers and decimals to any placeRounding is one of those essential basic skills that enhance number sense and mathematical reasoning. Rounding helps a student arrive at a friendly number, so it is easier to estimate probable outcomes, and rounding helps us understand larger more complex numbers better.1714500136525In creating your “Einstein Estimate” you just want to change numbers to easy friendly numbers for yourself. For instance: 278,642 ≈ 300,000 for me, because it is super easy, but some of you might change it to 280,000 or 270,000 based on other numbers being estimated, like 7 and 9.00In creating your “Einstein Estimate” you just want to change numbers to easy friendly numbers for yourself. For instance: 278,642 ≈ 300,000 for me, because it is super easy, but some of you might change it to 280,000 or 270,000 based on other numbers being estimated, like 7 and 9.0128905001714500261620With decimals, we often round long extended decimals to hundredths, because that mirrors our understanding of money. For instance: 3.26893245 would be rounded to 3.27, or if I’m doing an Einstein Estimate I’ll round it to a simple 3.00With decimals, we often round long extended decimals to hundredths, because that mirrors our understanding of money. For instance: 3.26893245 would be rounded to 3.27, or if I’m doing an Einstein Estimate I’ll round it to a simple 3.-114300157480The Procedures for Rounding: 1) Underline the digit you are asked to round too 2) Circle the digit on the right and draw an arrow over to the underlined digit, and ask yourself, “Does the circled digit, remember 5 is the magic number, change the underlined digit?” 3) The underlined digit will either stay the same or move up by one, and if the circled digit is ineffective in changing the underlined digit, then cross it out.00The Procedures for Rounding: 1) Underline the digit you are asked to round too 2) Circle the digit on the right and draw an arrow over to the underlined digit, and ask yourself, “Does the circled digit, remember 5 is the magic number, change the underlined digit?” 3) The underlined digit will either stay the same or move up by one, and if the circled digit is ineffective in changing the underlined digit, then cross it out.-114300-2540EXAMPLE00EXAMPLERound 27.36 to the nearest tenth.106172080010001257300121920002 7 . 3 6 2 7 . 4135920272043900Round 816.243 to the nearest hundredth.155448018224500144018013335008 1 6 . 2 4 3 8 1 6 . 2 4142778272044000Round 724.398 to the nearest hundredth.162306018224500 7 2 4 . 3 9 8 7 2 4 . 4 0594360039370BASE TEN0BASE TEN228600-228600ADDING and SUBTRACTING MULTI-DIGIT DECIMALS00ADDING and SUBTRACTING MULTI-DIGIT DECIMALS5372100-342900Eureka Math: Mod. 1 L. 9-10MATH CCSS: 5.NBT.B.70Eureka Math: Mod. 1 L. 9-10MATH CCSS: 5.NBT.B.7Standard: Students will be able to add and subtract decimals to hundredths using models, drawings, properties of operations, and/or the relationship between addition and subtraction. Students will also be able to write a written explanation of reasoning.The procedure for adding multi-digit (many numbered) decimals is to line up the decimal points. This means you need to stack your addends or subtrahends vertically, and the decimal points have to be directly on top of each other.11430080010In class I ask: “What is the procedure for the adding and subtracting of decimals?”00In class I ask: “What is the procedure for the adding and subtracting of decimals?”388620080010And you all reply: “Line up the decimals. Chop.” while moving your paw up and down grizzly-style. Kinesthetically, this helps you memorize the procedure.00And you all reply: “Line up the decimals. Chop.” while moving your paw up and down grizzly-style. Kinesthetically, this helps you memorize the procedure.21717002540000-11430041275EXAMPLE00EXAMPLE240030046355Adding with zeroes is no problem, but subtracting is a whole different issuebecause of borrowing and regrouping.0Adding with zeroes is no problem, but subtracting is a whole different issuebecause of borrowing and regrouping.42.1 + 1.235216.5 – 0.73251435002495550053721002495550056007002495550054864002495550042.100216.500 + 01.235 - 000.732 43.335215.7682857500235585Look at these examples without the zeroes and the decimals lined up.0Look at these examples without the zeroes and the decimals lined up.0.32167 + 21,712.4 12.24 – 11.642945319126207556500 0.3216712.24388532117991800 + 21712.4 -11.6429453 50292001885950052578001885950053721001885950054864001885950056007001885950057150001885950058293001885950059436001885950000000.3216712.240000019843758255000 + 21712.40000 -11.6429453 914400283845Notice that I dropped the comma, that’s so Idon’t get confused with the decimal point.0Notice that I dropped the comma, that’s so Idon’t get confused with the decimal point. 21712.72167 0.5970547 5943600114935BASE TEN0BASE TEN5372100-457200Eureka Math: Mod. 1 L. 11 & 12MATH CCSS: 5.NBT.B.5 & 70Eureka Math: Mod. 1 L. 11 & 12MATH CCSS: 5.NBT.B.5 & 7342900-228600MULTIPLYING MULTI-DIGIT DECIMALS00MULTIPLYING MULTI-DIGIT DECIMALSStandard: Fluently multiply multi-digit whole numbers using the standard algorithm, and multiply decimals to the hundredths using concrete models or drawings and strategies. Be able to explain and reason your methodology.The procedure for multiplying multi-digit decimals is to apply the number of spaces behind the decimal point in each factor (the problem) to the number of spaces in the product (the answer) coming in from the right. 114300110490In class I will ask you: “What is the procedure for the multiplication of decimals?”00In class I will ask you: “What is the procedure for the multiplication of decimals?”2628900110490And you all will reply kinesthetically: “The number of spaces in the problem equal the number of spaces in the answer.”0And you all will reply kinesthetically: “The number of spaces in the problem equal the number of spaces in the answer.”092710003543300201930Mr. Miller’s Method: 1 . 2 1 × 0 . 0 4 3 3 6 3 + 4 8 4 0. 0 5 2 0 3Mr. Miller’s Method: 1 . 2 1 × 0 . 0 4 3 3 6 3 + 4 8 4 0. 0 5 2 0 35257800255270OUT0OUTAn explanation from the 53999281869620059436002006605SPACES005SPACESKhan Academy on the 540050912073500502920014605000468630014605000MATHEMATICAL REASONING:125730086360? 1000? 100 5029200641350043434006413500468630064135001257300173355? 10000? 100011430005905500 1.21 1211143000444500 × 0.043 × 43528828013398500 3631143000183515÷ 100 ÷ 1000÷ 100 ÷ 1000 + 4840605790014605IN0IN8001006921500 .05203 5203365760020320004000500203200043434002032000468630020320005029200203200059436003078480BASE TEN0BASE TEN-114300678180When we are working with decimals, it is imperative that we understand the con-cept of place value. That is one reasonwhy we often refer to money when speaking with decimals. Imagine a dollar broken up into tenths (0.1, or dimes) or hundredths (0.01 or pennies), but we’d never break up a dollar into thousandths (0.001) or ten thousandths (0.0001). That would be the same as if we cut each penny into ten equal pieces (thousandths) or a hundred equal pieces (ten thousandths).00When we are working with decimals, it is imperative that we understand the con-cept of place value. That is one reasonwhy we often refer to money when speaking with decimals. Imagine a dollar broken up into tenths (0.1, or dimes) or hundredths (0.01 or pennies), but we’d never break up a dollar into thousandths (0.001) or ten thousandths (0.0001). That would be the same as if we cut each penny into ten equal pieces (thousandths) or a hundred equal pieces (ten thousandths).-114300106680HereHomershows theproceduralmovementformultiplyingdecimals.00HereHomershows theproceduralmovementformultiplyingdecimals.1028700301625005372100-342900Eureka Math: Mod. 1 L. 12MATH CCSS: 5.NBT.B.70Eureka Math: Mod. 1 L. 12MATH CCSS: 5.NBT.B.7342900-114300CREATING A FRIENDLY NUMBER00CREATING A FRIENDLY NUMBERStandard: Students should be able to add, subtract, multiply, and divide decimals to hundredths, and one way to double check their work is to change these seemingly difficult numbers to more friendly numbers and follow the operation.Students should be able to take any number and change it to a more friendly number and compute the operation. This will help us estimate an answer, which becomes increasingly more helpful as we work through the operations of adding decimals, subtracting, multiplying, and dividing decimals. 53721006350The “key” to the estimate is looking at numbers in an easy way.00The “key” to the estimate is looking at numbers in an easy way.37719009144000When estimating an operation, ( +, ?, ×, or ÷ ), we are basically changing large difficult numbers into easy “friendly” numbers to compute quickly an estimate. Mr. Miller calls this the “Einstein Estimate.”24.489 ÷ 4.16 =53721007620Don’t over think an easy solution to the problem.00Don’t over think an easy solution to the problem.This looks like a lot of work at first, but if I round first then it seemingly got a lot easier:365760052705EXAMPLES00EXAMPLES24 ÷ 4 = 63657600159385Remember the gas station sign telling us how much gas is:Some people will see regular unleaded gas at $1.73, but its really closer to $1.74. Is $1.74 friendly? Not really. $1.80 is morefriendly while $2 is the friendliest, but $1.73 9/10 is a ridiculous number to try and use.0Remember the gas station sign telling us how much gas is:Some people will see regular unleaded gas at $1.73, but its really closer to $1.74. Is $1.74 friendly? Not really. $1.80 is morefriendly while $2 is the friendliest, but $1.73 9/10 is a ridiculous number to try and use.571500055626000Your estimate should take no longer than 15 seconds EVER. The above problem should have just taken a few seconds. The real answer is 5.886, which is pretty close to 6.LOOKING AT IT ANOTHER WAYWhat if 24.489 ÷ 5.42, now 5 doesn’t really go into 24 easily or quickly. In other words the number 5 is not a factor for the product 24. Here are some ideas to consider:1) I could still change the 5.42 to a 4, because 4 is a factor of 24 and my quotient would be 6.365760040640 NOTICE HOW I BREAK A LOT OF ROUNDING RULES TO ESTIMATE HERE QUICKLY0 NOTICE HOW I BREAK A LOT OF ROUNDING RULES TO ESTIMATE HERE QUICKLY3344216167240002) I could change my divisor from 5.42 to a 6, because 6 is a factor of 24. That would make a quotient of 43657600787406.08 ? 18.4 ≈ 6 ? 18 ≈ 6 ? 20Which is friendlier? 18 or 20? Well I know 6 ? 9 is 54, so doubled would be 108. 6 ? 10 is 60; doubled is 120.06.08 ? 18.4 ≈ 6 ? 18 ≈ 6 ? 20Which is friendlier? 18 or 20? Well I know 6 ? 9 is 54, so doubled would be 108. 6 ? 10 is 60; doubled is 120.3) Lastly, I could change the dividend, 24.489 to 25, because I know 5 is a factor of 25, so the quotient would be 5.6057900406400BASE TEN0BASE TENThe quotient to 24.489 ÷ 5.42 = 4.518. All of my estimates are reasonably close to the actual quotient.228600-228600DIVIDING DECIMALS IN THE DIVIDEND00DIVIDING DECIMALS IN THE DIVIDEND5372100-342900Eureka Math: Mod. 1 L. 13-16MATH CCSS: 5.NBT.B.6 & 70Eureka Math: Mod. 1 L. 13-16MATH CCSS: 5.NBT.B.6 & 7Standard: Find whole number quotients of whole numbers with four-digit dividends and two-digit divisors using the properties of operations. Students should be able to illustrate and explain the calculation by using equations, arrays, and/or area models. Students also need to be able to divide decimals to hundredths.A solo decimal point in the dividend requires a student to put the decimal point directly on top of the house; of course, ensuring that place value is maintained throughout the division algorithm.-11430033020EXAMPLES00EXAMPLES262890055245This is the whole procedure.0This is the whole procedure.375.15 ÷ 15 2,898.736 ÷ 233883025120650013885555520700 . .4800600571500685800571500 15 375.1523 2898.732700383206068E.E..00E.E..2628900-571500285750053975 2020 4000 2020 400 25.01 126.0324800600-635003086100-63500685800-63500207835588900015 375.1523 2898.736 - 30 - 2337317241794700262890016256000 75 59269691962184E.E..00E.E..2857500222250 15020 30000 15020 3000 - 75 - 46308610016764000 0 15 138- 15 - 138 0 0 733543300172720THOUSANDTHS00THOUSANDTHS3086100172720HUNDREDTHS00HUNDREDTHS2628900172720TENTHS00TENTHS2171700172720DECIMAL PT.00DECIMAL PT.1714500172720ONES00ONES1257300172720TENS00TENS800100172720HUNDREDS00HUNDREDS342900172720THOUSANDS00THOUSANDS22860058420 1 2 6 . 0 3 2 23 2 8 9 8 . 7 3 60 1 2 6 . 0 3 2 23 2 8 9 8 . 7 3 6 - 69 46- 46285750069215- - - - - - - - - - - - - - - - 00- - - - - - - - - - - - - - - - 240030069215- - - - - - - - - - - - - - - - 00- - - - - - - - - - - - - - - - 194310069215- - - - - - - - - - - - - - - - 00- - - - - - - - - - - - - - - - 148590069215- - - - - - - - - - - - - - - - 00- - - - - - - - - - - - - - - - 102870069215- - - - - - - - - - - - - - - - 00- - - - - - - - - - - - - - - - 57150069215- - - - - - - - - - - - - - - - 00- - - - - - - - - - - - - - - - 11430069215- - - - - - - - - - - - - - - - 00- - - - - - - - - - - - - - - - 05029200226060This is how well you should have a division problem lined up.This is how well you should have a division problem lined up.457200243205004498340-254000594360067310BASE TEN0BASE TEN342900-228600DIVISION USING PLACE VALUE UNDERSTANDING00DIVISION USING PLACE VALUE UNDERSTANDING5143500-228600Eureka Math: Mod. 1 L. 14MATH CCSS: 5.NBT.B.70Eureka Math: Mod. 1 L. 14MATH CCSS: 5.NBT.B.7Standard: Students will be able to not only use the algorithm in division but relate the strategy to a written method and explain their reasoning.0387350EXAMPLES00EXAMPLESStudents should be able to use a place value chart to explain the division process using discs and place value to explain their reasoning in division.182880025400Starting this process is a little confusing, but the more you do it the easier it will get.Starting this process is a little confusing, but the more you do it the easier it will get.8.736 ÷ 31485900134620?0?800100-127000OnesTenthsHundredthsThousandths57150010160000342900101600000101600000101600004572002108200022860021082000021082000127169526390001275080120777000480060844550002286084455000994410558800004876808890000030480889000008001021590000800101016000012407901207770009601205588000061722010160000388620101600001600201016000016002010160000104013055880000582930444500001143044450000582930444500002400304445000058293010160000354330101600001257301016000012573044450000125730101600004572001797050011430017970500651510294005004229102940050019431029400500-34290294005008801106540500651510654050042291065405001943106540500-34290654050016002017970500582930179705002400301797050045720010477500114300104775006515102190750042291021907500880110-952500651510-952500422910-95250019431021907500-3429021907500194310-952500-34290-9525001600201047750058293010477500240030104775004572001441450011430014414500651510258445004229102584450019431025844500-34290258445008801102984500651510298450042291029845001943102984500-3429029845001600201441450058293014414500240030144145005372100183515Step 5I have nothing to carry over to the next column, so I scratch out a disc in the hundredths and make an even group. I do the same with thousandths.00Step 5I have nothing to carry over to the next column, so I scratch out a disc in the hundredths and make an even group. I do the same with thousandths.4000500183515Step 4I now have 27 discs in the tenths column. Well 27 ÷ 3 = 9. Literally I scratch out a disc and place it in the row below.0Step 4I now have 27 discs in the tenths column. Well 27 ÷ 3 = 9. Literally I scratch out a disc and place it in the row below.262890010795Step 3The goal is to now to begin with the 8 discs in the ones column and divide them evenly among the three rows below.0Step 3The goal is to now to begin with the 8 discs in the ones column and divide them evenly among the three rows below.125730020955Step 2Based on the value of the digit in the dividend, I am going to place a number of discs in that box.0Step 2Based on the value of the digit in the dividend, I am going to place a number of discs in that box.-11430020955Step 1Based on the divisor, 3, I will create three rows below the placement of my discs.0Step 1Based on the divisor, 3, I will create three rows below the placement of my discs.-11430079375Explanation: I first placed 8 discs in the ones column, 7 in the tenths, 3 in the hundredths and 6 in the thousandths. Because the divisor is 3 I created three rows below. I found that I could make a group of 2 in the ones with 2 left over, which meant I transferred a total of 20 discs into the tenths column (10 discs for each one). Next, I could make a group of 9 in the tenths column with nothing left over, a group of I in the hundredths and lastly a group of 2 in the thousandths. My answer was 2.912.00Explanation: I first placed 8 discs in the ones column, 7 in the tenths, 3 in the hundredths and 6 in the thousandths. Because the divisor is 3 I created three rows below. I found that I could make a group of 2 in the ones with 2 left over, which meant I transferred a total of 20 discs into the tenths column (10 discs for each one). Next, I could make a group of 9 in the tenths column with nothing left over, a group of I in the hundredths and lastly a group of 2 in the thousandths. My answer was 2.912.392430017773650037719001624965005943600258445BASE TEN0BASE TEN3924300982345005257800-228600Eureka Math: Mod. 1 L. MATH CCSS: 5.NBT.B.70Eureka Math: Mod. 1 L. MATH CCSS: 5.NBT.B.7228600-114300DIVIDING DECIMALS WITH A DECIMAL IN THE DIVISOR AND DIVIDEND00DIVIDING DECIMALS WITH A DECIMAL IN THE DIVISOR AND DIVIDENDStandard: The lesson below is really more of a 6th Grade example, meaning that the dividend and divisor are a little larger than expected for 5th Graders, but the quotient is also a decimal. For 5th Graders the quotient will always be a whole number.The decimal point in the divisor dictates how much the decimal point moves to the right in the dividend. In other words, whatever number of spaces you move the decimal point in the divisor, you must also move the decimal point in the dividend… even if there isn’t a decimal point in the dividend. When I ask students, “What is the procedure for the division of decimals?” They reply, “The number of spaces in the divisor equal the number of spaces in the dividend… to the right.”031750EXAMPLES00EXAMPLES999.6 ÷ 58.89.4806 ÷ 6.7337925381890910025408621836660050292002044700091440020447000 . . 29718000This is the whole procedure.This is the whole procedure.80010022860000182880022860000572135023939500549334223588000468630022860000491490022860000 5 8.8 9 9 9.6 . 6.7 3 9.4 8 .0 691440023368000502920023368000 1 7 . 1 .4 0 8 57150002228850054864002228850049149002228850046863002228850018288002228850080010022288500 5 8.8 9 9 9.6 . 6.7 3 9.4 8 .0 6 0 - 5 8 8 - 6 7 3 4 1 1 6 2 7 5 0 - 4 1 1 6 - 2 6 9 2342900070485EXPLANATIONIn this example you are multiplying the divisor, 6.73 by 100, and the dividend by 100 to maintain balance. So 6.73 becomes 673 and 1.4806 becomes 148.06.00EXPLANATIONIn this example you are multiplying the divisor, 6.73 by 100, and the dividend by 100 to maintain balance. So 6.73 becomes 673 and 1.4806 becomes 148.06. 0 5 8 6 0- 5 3 8 4114300114935EXPLANATIONYou are basically multiplying the divisor, 58.8 by 10, which equals 588. If you multiply the divisor by 10, then you must also multiply the dividend by 10 to maintain balance. So 999.6 becomes 9996.00EXPLANATIONYou are basically multiplying the divisor, 58.8 by 10, which equals 588. If you multiply the divisor by 10, then you must also multiply the dividend by 10 to maintain balance. So 999.6 becomes 9996. 4 7 65943600125095BASE TEN0BASE TEN5372100114300MATH CCSS: 5.NF.B.3MATH CCSS: 6.RPA.3.C0MATH CCSS: 5.NF.B.3MATH CCSS: 6.RPA.3.C5372100-342900Eureka Math: Mod. 1 L. MATH CCSS: 4.NF.C.6 & 70Eureka Math: Mod. 1 L. MATH CCSS: 4.NF.C.6 & 7114300-2286005TH GRADE CONVERTING FRACTIONS TO DECIMALS AND DECIMALS TO FRACTIONS005TH GRADE CONVERTING FRACTIONS TO DECIMALS AND DECIMALS TO FRACTIONS1143001739906TH GRADE CONVERTING TO PERCENTS006TH GRADE CONVERTING TO PERCENTSStandard: In 4th Grade students learned equivalency of fractions and that they could convert.54 to 54/100. In 5th Grade students learn that fractions are division of the numerator by the denominator, thus leading to a decimal equivalent. True conversion of fractions to decimals to percent is a 6th Grade standard.Convert Decimals to Fractions and PercentConverting a decimal to a fraction is as simple as identifying its place value. Take the decimal point away and draw in a line for the fraction bar underneath. That number is your numerator, and the denominator is its place value. .7 = 7 .32 = 32 0.375 = 375 10 100 1000Converting a decimal to a percent is as easy as removing the decimal point and adding a percentage sign. We are literally taking, .64, and multiplying it by 100 to get 64%.Converting Percent to Decimals and FractionsTo convert a percent to a decimal, first remove the percentage sign and add a decimal point from the right side coming in two places as in 42% is .42 or 630% is 6.30.5029200497840This mathematical practice is the piece that is more of a 6th Grade standard.This mathematical practice is the piece that is more of a 6th Grade standard.3543300558165SIMPLIFYING00SIMPLIFYINGWith converting a percent to a fraction go one step more by then following the steps in converting a decimal to a fraction as in this example: 64% = .64 = 64 ÷ 2 = 32 ÷ 2 = 16 100 ÷ 2 50 ÷ 2 25Converting Fractions to Decimals and PercentTo convert a fraction to a decimal, simply divide by locking down the divisor and moving the numerator into the dividend position. Add a decimal point and three zeroes.6057900544830BASE TEN0BASE TENAnd in converting a fraction to a percent, do step one above, round it to the nearest hundredth, then multiply that decimal by 100. That will be your percent.5th GRADE SKILLS I SHOULD BE ABLE TO DOIN NUMBERS AND OPERATIONS IN BASE TEN1) Recreate a place value chart accurately.2) Understand that a digit in the hundreds column is 10 times larger than a digit in the tens column and 1/10 as great as a digit in the thousands column.3) Understand the value of a digit given its place value4) Understand that every number has a decimal point whether you see it or not.5) Understand the “Powers of 10” and be able to multiply and divide using that understanding6) Be familiar with and use the metric system in measuring length and volume, and convert units within the system7) Be able to write out numbers in written form8) Be able to write out numbers in expanded form9) Be able to compare two numbers by stacking them and using the symbols: < and >10) Round any whole number or decimal to another place value11) Follow the procedures for rounding a number12) Use rounding to create “friendly” numbers for quick estimation13) Know and understand the procedure for adding and subtracting decimals14) Be able to add and subtract decimals fluently15) Know and understand the procedure for the multiplication of decimals16) Be able to multiply decimals fluently17) Know and understand the procedure for the division of decimals in the dividend18) Know and understand the procedure for the division of decimals in the divisor19) Be able to divide decimals in the dividend and divisor fluently.20) Be able to convert and understand fraction to decimal equivalency21) Be able to convert and understand decimal to fraction equivalency ................
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