6.4: Addition and Subtraction of Decimals - Mathematics

Learning Objectives

• understand the method used for adding and subtracting decimals
• be able to add and subtract decimals
• be able to use the calculator to add and subtract decimals

The Logic Behind the Method

Consider the sum of 4.37 and 3.22. Changing each decimal to a fraction, we have

(4 dfrac{37}{100} + 3 dfrac{22}{100}) Performing the addition, we get

(egin{array} {rcl} {4.37 + 3.22 = 4 dfrac{37}{100} + 3 dfrac{22}{100}} & = & {dfrac{4 cdot 100 + 37}{100} + dfrac{3 cdot 100 + 22}{100}} {} & = & {dfrac{437}{100} + dfrac{322}{100}} {} & = & {dfrac{437 + 322}{100}} {} & = & {dfrac{759}{100}} {} & = & {7 dfrac{59}{100}} {} & = & { ext{seven and fifty-nine hundredths}} {} & = & {7.59} end{array})

Thus, (4.37 + 3.22 = 7.59).

The Method of Adding and Subtracting Decimals

When writing the previous addition, we could have written the numbers in col­umns.

(egin{array} {r} {4.37} {underline{+3.22}} {7.59} end{array})

This agrees with our previous result. From this observation, we can suggest a method for adding and subtracting decimal numbers.

Method of Adding and Subtracting Decimals

Align the numbers vertically so that the decimal points line up under each other and the corresponding decimal positions are in the same column.
Add or subtract the numbers as if they were whole numbers.
Place a decimal point in the resulting sum or difference directly under the other decimal points.

Sample Set A

Find the following sums and differences.

(9.813 + 2.140)

Solution

(egin{array} {r} {9.813} {underline{+2.140}} {11.953} end{array}) The decimal points are aligned in the same column.

Sample Set A

(841.0056 + 47.016 + 19.058)

Solution

(egin{array} {r} {841.0056} {47.016 } {underline{+19.058 }} end{array})

To insure that the columns align properly, we can write a 0 in the position at the end of the numbers 47.016 and 19.058 without changing their values.

Sample Set A

(1.314 - 0.58)

Solution

(egin{array} {r} {1.314} {underline{-0.58 }} end{array}) Write a 0 in the thousandths position.

Sample Set A

(16.01 - 7.053)

Solution

(egin{array} {r} {16.01 } {underline{-7.053}} end{array}) Write a 0 in the thousandths position.

Sample Set A

Find the sum of 6.88106 and 3.5219 and round it to three decimal places.

Solution

(egin{array} {r} {6.88106} {underline{+3.5219 }} end{array}) Write a 0 in the ten thousandths position.

We need to round the sum to the thousandths position. Since the digit in the position immediately to the right is 9, and 9>5, we get

10.403

Sample Set A

Wendy has $643.12 in her checking account. She writes a check for$16.92. How much is her new account balance?

Solution

To find the new account balance, we need to find the difference between 643.12 and 16.92. We will subtract 16.92 from 643.12.

After writing a check for $16.92, Wendy now has a balance of$626.20 in her checking account.

Pracitce Set A

Find the following sums and differences.

(3.187 + 2.992)

6.179

Pracitce Set A

(14.987 - 5.341)

9.646

Pracitce Set A

(0.5261 + 1.0783)

1.6044

Pracitce Set A

(1.06 - 1.0535)

0.0065

Pracitce Set A

(16,521.07 + 9,256.15)

25,777.22

Pracitce Set A

Find the sum of 11.6128 and 14.07353, and round it to two decimal places.

25.69

Calculators

The calculator can be useful for finding sums and differences of decimal numbers. However, calculators with an eight-digit display cannot be used when working with decimal numbers that contain more than eight digits, or when the sum results in more than eight digits. In practice, an eight-place decimal will seldom be encoun­tered. There are some inexpensive calculators that can handle 13 decimal places.

Sample Set B

Use a calculator to find each sum or difference.

42.0638 + 126.551

Solution

 Display Reads Type 42.0638 42.0638 Press + 42.0638 Type 126.551 126.551 Press = 168.6148

The sum is 168.6148.

Sample Set B

Find the difference between 305.0627 and 14.29667.

Solution

 Display Reads Type 305.0627 305.0627 Press - 305.0627 Type 14.29667 14.29667 Press = 290.76603

The difference is 290.76603

Sample Set B

51.07 + 3,891.001786

Solution

Since 3,891.001786 contains more than eight digits, we will be unable to use an eight-digit display calculator to perform this addition. We can, however, find the sum by hand.

(egin{array} {r} {51.070000} {underline{3891.001786}} {3942.071786} end{array})

The sum is 3,942.071786.

Practice Set B

Use a calculator to perform each operation.

(4.286 + 8.97)

13.256

Practice Set B

(452.0092 - 392.558)

59.4512

Practice Set B

Find the sum of 0.095 and 0.001862

0.096862

Practice Set B

Find the difference between 0.5 and 0.025

0.475

Practice Set B

Find the sum of 2,776.00019 and 2,009.00012.

Since each number contains more than eight digits, using some calculators may not be helpful. Adding these by “hand technology,” we get 4,785.00031

Exercises

For the following 15 problems, perform each addition or subtraction. Use a calculator to check each result.

Exercise (PageIndex{1})

(1.84 + 7.11)

8.95

Exercise (PageIndex{2})

(15.015 - 6.527)

Exercise (PageIndex{3})

(11.842 + 28.004)

39.846

Exercise (PageIndex{4})

(3.16 - 2.52)

Exercise (PageIndex{5})

(3.55267 + 8.19664)

Exercise (PageIndex{6})

(0.9162 - 0.0872)

Exercise (PageIndex{7})

(65.512 - 8.3005)

57.2115

Exercise (PageIndex{8})

(761.0808 - 53.198)

Exercise (PageIndex{9})

(4.305 + 2.119 - 3.817)

2.607

Exercise (PageIndex{10})

(19.1161 + 27.8014 + 39.3161)

Exercise (PageIndex{11})

(0.41276 - 0.0018 - 0.00011)

0.41085

Exercise (PageIndex{12})

(2.181 + 6.05 + 1.167 + 8.101)

Exercise (PageIndex{13})

(1.0031 + 6.013106 + 0.00018 + 0.0092 + 2.11)

9.135586

Exercise (PageIndex{14})

(27 + 42 + 9.16 - 0.1761 + 81.6)

Exercise (PageIndex{15})

(10.28 + 11.111 + 0.86 + 5.1)

27.351

For the following 10 problems, solve as directed. A calculator may be useful.

Exercise (PageIndex{16})

Add 6.1121 and 4.916 and round to 2 decimal places.

Exercise (PageIndex{17})

Add 21.66418 and 18.00184 and round to 4 decimal places.

39.6660

Exercise (PageIndex{18})

Subtract 5.2121 from 9.6341 and round to 1 decimal place.

Exercise (PageIndex{19})

Subtract 0.918 from 12.006 and round to 2 decimal places.

11.09

Exercise (PageIndex{20})

Subtract 7.01884 from the sum of 13.11848 and 2.108 and round to 4 decimal places.

Exercise (PageIndex{21})

A checking account has a balance of $42.51. A check is written for$19.28. What is the new balance?

$23.23 Exercise (PageIndex{22}) A checking account has a balance of$82.97. One check is written for $6.49 and another for$39.95. What is the new balance?

Exercise (PageIndex{23})

A person buys $4.29 worth of hamburger and pays for it with a$10 bill. How much change does this person get?

$5.71 Exercise (PageIndex{24}) A man buys$6.43 worth of stationary and pays for it with a $20 bill. After receiving his change, he realizes he forgot to buy a pen. If the total price of the pen is$2.12, and he buys it, how much of the $20 bill is left? Exercise (PageIndex{25}) A woman starts recording a movie on her video cassette recorder with the tape counter set at 21.93. The movie runs 847.44 tape counter units. What is the final tape counter reading? Answer 869.37 Exercises for Review Exercise (PageIndex{26}) Find the difference between 11,206 and 10,884. Exercise (PageIndex{27}) Find the product, (820 cdot 10,000). Answer 8,200,000 Exercise (PageIndex{28}) Find the value of (sqrt{121} - sqrt{25} + 8^2 + 16 div 2^2). Exercise (PageIndex{29}) Find the value of (8 dfrac{1}{3} cdot dfrac{36}{75} div 2 dfrac{2}{5}). Answer (dfrac{20}{9} = dfrac{5}{3}) or (2 dfrac{2}{9}) Exercise (PageIndex{30}) Round 1.08196 to the nearest hundredth. CBSE Class 7 Mathematics- Chapter 2- Fractions and Decimals- Addition and Subtraction of Decimals Notes. Write one number of the top of other, such that the decimal points line up vertically. CBSE Class 7 Mathematics Notes Addition and Subtraction of Decimals Add: 0.19 + 2.3 Decimal numbers, 0.19 and 2.3 have two digits and one digit respectively to the right of the decimal point. So, we add a zero to the right of 2.3. Subtract: 39.87 &ndash 21.98 Decimals numbers 39.87 and 21.98 have the same number of zeros after the decimal point. Example: Dinesh went from place A to place B and from there to place C. A is 7.5 km from B and B is 12.7 km from C. Ayub went from place A to place D and from there to place C. D is 9.3 km from A and C is 11.8 km from D. Who travelled more and by how much? Distance travelled by Dinesh = Distance from A to B + Distance from B to C =7.5 km + 12.7 km Distance travelled by Dinesh = 20.2 km Distance travelled by Ayub =Distance from A to D + Distance from D to C = 9.3 km + 11.8 km Distance travelled by Ayub = 21.1 km We see that the distance travelled by Ayub is more than the distance travelled by Dinesh. Mathematics - Class 4 / Grade 4 Step1. Change all decimal fractions/decimal numbers into like decimal fractions. Step2: Arrange all the decimal fractions in columns according to their place values in such a way that decimal points come in line. Step1: Change into like decimal fractions. Here, the greatest number of decimal places is two. 3.1 has 1 decimal place, we will add one 0 after 1 and so, 3.1 is written as 3.1 0 0.48 already has two decimal places. Step2: Arrange all the decimal fractions in columns according to their place values in such a way that decimal points come in line. Steps to Subtract Decimals Follow same steps as in addition Step1: Change into like decimal fractions. Here, the greatest number of decimal places is two. 8 has no decimal place, we will add two zero after putting decimal point so, 8 is written as 8.0 2.34 already has two decimal places. Step2: Arrange all the decimal fractions in columns according to their place values in such a way that decimal points come in line.(write greater decimal fraction on the top) Subtracting Decimals Example 1: A customer buys$6.33 of food in a store. If he pays with a $10 bill, then how much change should the cashier give him? Analysis: The cashier needs to subtract the two decimals in order to make change for the customer. Step 1: You must first line up the decimal points in a column. Step 2: Start on the right, and subtract each column in turn. Note that you are subtracting digits in the same place-value position. Step 3: If the digit you are subtracting is bigger than the digit you are subtracting from, you have to borrow a group of ten from the column to the left. Step 4: Be sure to place the decimal point in the difference. Answer: The cashier should give the customer$3.67 in change.

Example 2: Subtract: 8.06 - 8.019

Step 1: You must first line up the decimal points in a column.

Step 2: The decimals in this problem do not have the same number of decimal digits. You can write an extra zero to the right of the last digit of the first decimal so that both decimals have the same number of decimal digits.

Step 3: Start on the right, and subtract each column in turn. (Subtract digits in the same place-value position.)

Step 4: If the digit you are subtracting is bigger than the digit you are subtracting from, you have to borrow a group of ten from the column to the left.

Step 5: Be sure to place the decimal point in the difference.

In Example 1, we used four steps to subtract decimals. However, in Example 2, we had the extra step of writing an extra zero to the right of the last digit of the first decimal so that both decimals have the same number of decimal digits. Accordingly, we have summarized the 5-step procedure for subtracting decimals below.

Procedure: To subtract decimals, follow these steps:

1. Line up the decimal points in a column.
2. When needed, write one or more extra zeros to the right so that both decimals have the same number of decimal digits.
3. Start on the right, and subtract each column in turn. (Subtract digits in the same place-value position.)
4. If the digit you are subtracting is bigger than the digit you are subtracting from, you have to borrow a group of ten from the column to the left.
5. Place the decimal point in the difference.

Example 3: Two students were asked to subtract these numbers: $88 -$77.23. Student 1 calculated a difference of $11.23 and Student 2 calculated a difference of$10.77. Which student is correct? Explain your answer.

Analysis: Student 1 did not follow the procedure for subtracting decimals: He did not write extra zeros as place holders and he did not borrow. As a result, the difference calculated by Student 1 is incorrect. Student 2 followed the correct procedure and calculated the correct difference.

Answer: Student 2 is correct: $88 -$77.23 = 10.77 Example 4: Subtract: 32.5 - 7.94 Answer: 32.5 - 7.94 = 24.56 Example 5: Subtract: 30.041 - 9.785 Answer: 30.041 - 9.785 = 20.256 Example 6: Subtract: 18.2 - 3.199 Answer: 18.2 - 3.199 = 15.001 Example 7: In a 200-meter dash, the first-place winner reached the finish line in 19.75 s and the second-place winner reached the finish line in 19.8 s. What is the difference in their times? Answer: The difference in their times is 0.05 s. Summary: When subtracting decimals, first line up all the decimal points in a column. When needed, write one or more extra zeros to the right so that both decimals have the same number of decimal digits. Subtract digits in the same place-value position. When needed, borrow a group of ten from the column to the left. Place the decimal point in the difference. Exercises Directions: Read each question below. You may use paper and pencil to help you subtract. Click once in an ANSWER BOX and type in your answer then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Decimal Math Games Are you looking for free decimal math games? On this page you can find games for the following important topics: rounding decimals to the nearest whole number, estimating sums with decimals, comparing and ordering decimals, adding and subtracting decimals using tenths, hundredths, and thousandths, multiplying decimals, multiplying decimals with whole numbers, changing decimals to fractions and percents, and solving word problems with decimals. The math games on this page are designed to reinforce basic facts about decimals. Students all over the world love soccer. They will probably love these math soccer games about decimals. Adding Decimals Add tenths, hundredths, and thousandths with sums greater than one. Subtracting Decimals Soccer Fun soccer game about subtracting decimals. Rounding Decimals to the Nearest Whole Number Learn how to round decimals by playing this exciting game. How many homeruns can you hit? Play the following fast-paced baseball math games about decimals to show off your skills. Rounding Decimals Students can practice rounding decimals to different place values. Adding Decimals Add decimals and earn hundreds of points when playing this fun baseball game. Subtracting Decimals Baseball Subtract decimals in this fast-paced baseball math game. Decimals Board Game Check your knowledge with this online decimals board game and be the first one to get to the finish line. For each correct answer, you will be able to roll the die to move forward on the game board to get to the finish line. Do you love football? Try playing these fun football math games about decimals. Decimals Place Value Solve problems about place value in decimals in this interactive football math game. Play the following basketball math games alone or with a partner. These exciting games can also be played in the classroom by using a Promethean Board or an LCD projector and dividing the students into two teams. Adding Decimals Game Fun basketball math game about adding decimals. Subtracting Decimals Basketball Game Subtract decimals correctly to score as many points as possible. Multiplying Decimals Game Do yo know how to multiply decimals? Show off your knowledge and skills by playing this game. The following jeopardy games have a single-player mode, as well as a multi-player feature. You can play these games at home or at school as review activities. Decimals Jeopardy Game Review the four operations with decimals by playing this jeopardy game. Fractions, Decimals, Percets Jeopardy Game Change fractions to decimals and percents and vice-versa. Changing Fractions and Decimals to Percents Demonstrate your math skills and earn points by playing this interactive game about changing decimals to percents and vice-versa. Changing Fractions to Percents In this concentration game, students will match different fractions with the equivalent percents. Related Resources The various resources listed below are aligned to the same standard, (5NBT07) taken from the CCSM (Common Core Standards For Mathematics) as the Decimals Worksheet shown above. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction relate the strategy to a written method and explain the reasoning used. Example/Guidance Summary Worksheet e.g. 4.234 - 3.438 e.g. 6.892 - 3.2 e.g. .4 x .6 e.g. .44 x 7.3 e.g. 6.004 x 100 e.g. 5.587 x .65 e.g. 3.67 ÷ 7 e.g. 86 ÷ .007 e.g. 86 ÷ .007 e.g. 3.563 + 6.451 e.g. 3.754 + 2.1 Worksheet Generator Similar to the above listing, the resources below are aligned to related standards in the Common Core For Mathematics that together support the following learning outcome: Perform operations with multi-digit whole numbers and with decimals to hundredths Steps to add or subtract Decimals: Convert decimals to like decimals. (Decimals that have the same number of digits after the decimal point are like decimals). Write the decimals one below the other as per the places of the digits. Add or subtract starting from the rightmost digit and moving towards the leftmost digit. Place the decimal point under the decimal point in the answer. Example: Add 23.45 13.101 and 345.5 1. Convert to like decimals: The highest decimal place is 3, so we add zeros in other numbers and get 3 decimal places in them too. 2. Line up the decimals: 4. Place decimal in answer: Example: Kylie had 25 m of ribbon. She uses 8 m and 13 cm to decorate a skirt. How much ribbon is remaining with Kylie? Length of ribbon Kylie had = 25 m = 25.00 Length of ribbon used by Kylie = 8 m 13 cm = 8.13 The remaining length of ribbon = 25.00 &ndash 8.13 The word &ldquodecimal&rdquo originated in the early 17th century from the Latin word &ldquodecimus&rdquo which means &ldquotenth&rdquo. Decimal Arithmetic Subtracting one decimal from the other is similar to addition. Two positive decimal numbers may be subtracted as follows: • the numbers are written one below the other with the decimal points vertically aligned • mentally place a zero in any places where only one of the numbers has a digit. • the numbers are subtracted as if they are whole numbers • a decimal point is placed in the difference such that it is directly below the decimal points in the numbers being subtracted To begin with given are examples where the numbers being subtracted are positive and the answer will be positive. Example Let us calculate: 123.45 - 37.5 Example Exercise Perform the following subtractions without using a calculator: Small positive number minus large positive number Subtracting a large number from a small number will always give a negative number, the opposite of subracting the small number from the large number. This can be written: a &minus b = &minus(b &minus a) Example Calculate 11.6 &minus 18.0 without using a calcuator. This is the opposite of 18.0 &minus 11.6 which equals 6.4. Exercise Perform the following subtractions without using a calculator: Subtracting Signed Decimal Numbers This section assumes you are already familiar with subtraction for integers. The things to remember are summarised here. • Subtraction of a number is equivalent to adding its negative. That is: x &minus y = x + (&minusy) for any numbers x and y. • Subtracting a negative number is equivalent to adding its positive. That is, x &minus (&minusy) = x + y Examples We show application of these rules to signed decimal numbers. Exercise Try these exercises with signed decimal numbers without using a calculator. 6.4: Addition and Subtraction of Decimals - Mathematics (These question links will help get you to some good advice, but please explore as many of the questions as you can. this page is loaded with good advice and activities!) > My children usually use their fingers to find the sum, but > when they get problems like 6 + 7, etc. and run out of > fingers they get so confused. How do you teach your > children to find sums in those situations where they run > out of fingers? Any cute tricks? Also with subtraction, > if they have to find the difference using numbers greater > than 10, they also get confused and say "I am out of > fingers." Any tricks for this? Thanks > new to first counting up and counting back For the addition facts, I tell the kids to "think the biggest number first and add up." Think "9" and count on 7 more for 7 + 9. For the subtraction facts, if there is a big number on the bottom, ex. 11 - 8, think 8 and count up to 11. Little number on the bottom, ex. 11 - 3, think 11 and count back 3. It works for us. Hope this helps. 2nd grade on teachers.net primary elementary board North Carolina Math Site This is a superb site with very readable, do-able math lessons and games--a lot that will help you with the drill and repetition that they will need, in a fun way. Scroll down to the First Grade and the year is divided into 4 quarters plus a section of blackline masters. In our school, to do these harder addition problems, we tell the students to "put the big number in your brain" and "count on" from that number using their fingers for the second addend. (eg) For 8 + 6, they would touch their head and say, "8 in my brain. (and putting up six fingers) 9, 10, 11, 12, 13, 14." This works well for my first grade. Darcy, on teachers.net primary elementary board If you have trouble getting to that great N.C. math instruction site, just type in Click on instructional resources. Under Assessment resources you will click on Strategies for Math Instruction. Good luck! It is worth the trouble to get there! Cindy, on teachers.net primary elementary board I really like the way Saxon math teaches the facts. They teach "strategies" instead of using fingers/manipulatives (although manipulatives are used in intros. and as needed). There are doubles, which the children simply memorize easily. There are "plus one" facts (like counting to the next number), "plus zero" facts, which, again, are easy to see after using manipulatives for a little bit). There are "doubles plus one" facts, "adding two" facts, etc. You teach a STRATEGY for remembering and then the kids practice DAILY in a variety of ways--paper practice, raps, string "wrap-ups", etc., etc. It's the best way I've seen so far, as opposed to the old methods we used to teach! > Does anyone have ideas on how to teach beginning addition > in first grade?? I am a first year teacher in first grade > and would appreciate any ideas. Thank You, > Kim :), pushing together sets of teddy bear counters I've found it really helpful to have my students use teddy bears to form two sets and then slide them together to find the sum. I start the lesson by telling them the number of bears they should put in each set. I make sure they place the first set on the left side of their desk and the second set on the right side. I then say the numbers for example " 5 plus 2 equals" When I say "equals" the students slide the sets together and then count to find the sum. I do this several times before writing a math sentence on the board. I stress the words "sets" and "sums". After we do this for a couple of days, then I allow my students to use their math workbook pages. I feel strongly that they children should "feel" the sets going together. I've found it to be very successful over the years. Glenda, on teachers.net primary elementary board I do an activity with my 1st graders called "Shake the Beans". You need the following: 1-2 bag(s) of large lima beans 1 can of green spray paint 1 can of orange spray paint 1 box of quart sized ziploc bags Spray the beans with green on one side and orange on the other side. After they are dry, put about 20-30 beans in each ziploc bag. Make enough bags so each student has his/her own bag. 1. Each student takes out a set number of beans that you are working on. For example, if you are wanting to find all the number combinations for 7, they would take 7 beans (only) out of their bag. Then zip the bag up and put in their desks or just somewhere away from where they are working. 2. Students will shake the beans gently in their hands and then gently let them out on their desk. 3. They put the green together on one side of their desk and the orange together on the other side of their desk. 4. Next, they are to record their results. There are several ways they can do this. They can make a table to record their info. For example, Green. 2 6 Orange 5 1 Total. 7 7 etc. They can draw pictures of the beans. They would draw the number of green beans with a green crayon and then the number of orange beans with an orange crayon. Then they can write the number under each set. I also have a worksheet that has beans drawn on them. They continue shaking the beans to find all the number combinations for the sum they are working on. Another great resource is the math series called Investigations - http://www.terc.edu/investigations/index.html The first two books in the first grade series are wonderful and filled with great hands-on activites to develop number sense. > Help!! I'm a Grad Student in NY and I need to devise a floor > game for basic facts. The content is not the problem for > me, it's the game. Everything I've come up with thus far > seems to get too complicated. Does anyone know of any good > sites for floor games (doesn't have to be math related) or > have any good ideas? > Laura, How about subtraction bowling? Make pins from 2 liter soda bottles. Students take turns bowling and writing the subtraction equation. 9 pins, 4 hit: 9 - 4 = 5 Give each bowler 2 turns. So the second turn would be 5 pins, 2 hit: 5 - 2 = 3 or you could reverse the system and have them add to find the total pins toppled in two tries. Make giant cards 1 - 12 Make giant dice Have students lay cards in a row 1 - 12 First student rolls the dice --- rolls a 4 and a 2 That student can turn over any combination of cards that equals 6 (5 and 1, 3 and 2 and 1, 2 and 4, . ) The first student plays until they turn over all the large cards. The turn ends if they can not turn over cards to equal the rolled amount. The next student takes their turn. The winner is the student who can turn over all the cards during their turn. Print this child friendly chart. Buy a box of Cheerio-sized cereal for math class and then have a blast. I use this blank with a page of problems. The kids use the top row to do addition and the bottom to do subtraction, and the fun part is when they do the take away part on the bottom they get to eat the cereal. Please try this. I just made it. Also check out the whole Run the Seasons program. You can find it at www.carolgoodrow.com It's going to be full of hands-on first grade academics. > I'm looking for a lesson for third grade that involves > addition and some form of sports, ie. baseball, football.. > Any help would be appreciated. It is for an observation. > Pat This is a game the kids beg for. Draw a field on the board. Just a line with 10 yards marked off down from the 50 towards both goals in the center and goals on each end. I have a magnetic whiteboard and made a football with a magnet on the back. It starts at the 50 yard line. The class is in two teams. The first two come up. I use flashcards but you could give problems etc. The first correct answer advances their ball 10 yards toward their goal. The ball will move back and forth. When a team gets to their goal, they get a touchdown. I have added some rules, such as: As the ref, I can give penalties for Unneccessary Roughness (pushing in lines), Interference (giving an answer when not your turn), Unsportsmanlike Conduct (rude to a teammate for not getting answer), Excessive Celebrating (that is not what it is called but when the player brags and showboats about beating someone). Adapt for you but it is fun. > Please help. Have tried everything I know. but still have > about 4 to 5 kids who cannot grasp subtraction. Have tried > manipulatives, number line, counting on fingers, physically > demonstrating, etc. nothing seems to be working. I know > they are getting frustrated, because I am as well. Thanks > for any help. Someone in chat said that they have probably > not internalized the numbers yet. Well, if that is the > case, I will need to seek out activities to promote such > learning. Thanks, > Lucy keep it simple and have it match their world: stories and manipulatives Lucy, I always start with stories that I make out loud. They can understand these because they match their real world. For example: Susie has 3 cute dollies that she keeps on her bed. Her little sister likes them too.One day her little sister just "borrows" one. When Susie comes home from school, how many are still on her bed? Or Tommy took 5 cookies to school for a snack, but he dropped one on the way to get on the bus. How many cookies does he have for snack now? Please try this. It works. EVERY kid, no matter how poor in math, will be able to answer these. I do this every year. Next, I start to get a little abstract. I let them use manipulatives to act these type of problems out. You can do this whole class. Another level will be writing the number sentence on the board, etc. They need to learn that the first number is the number you start with and the second is the number you take away. THEN you have to start getting a bit abstract. Start with a problem and see if they can use manipulatives to solve. Cereal works well, because you can put it out and then eat. I have some charts on KidsRunning that you can use. Go to http://www.kidsrunning.com/school/krschool022seasons.html Later on you'll teach other strategies, such as the counting back, counting up, difference, etc., but you need to start with the first couple of basic take aways to give a sound understanding to first graders. Good luck, Carol, KidsRunning.Com, on teachers.net primary elementary board Try the shell game. ex. I have a child count 5 manipulatives. Then I put three under a bowl (shell) and ask them how many I should put under the second bowl to make 5. They can see the remaining shells so that they should know the answer. I always give them the number sentence right after (5 - 3 = 2). I work from there until they can get the answer without having to see the remaining objects first. They love it when I lift the second bowl and they are right. The rest of the class can play this game with partners while you work with a small group. If they grasp the concept then you can proceed to abstract. Eileen G, on teachers.net primary elementary board Missing Addends, Cuisenaire Rods, "trash can" Wow. You have tried all the things that *should* work! I am assuming since you are not at your wit's end about addition that these children have absorbed the concept of addition successfully. So maybe you can turn that to their advantage. I have noticed that many children who can instantly tell me that 6 + 1 = 7 are thrown for a loop when I ask them "What is 7 - 6?" or even "What is 7 - 1?" When I turn it into an addition problem with a missing addend, sometimes that helps. For instance, "I'm holding up six fingers. How many more do I need to make 7?" They can usually tell me that I just need one more. Then I model for them a way that could be represented: 6 + ? = 7 Now, working backwards with the same problem: "I have 7 fingers up. If I put one down, how many will I have?" Since they have just modeled 6 + 1 = 7, it shouldn't be too hard to model 7 - 1 = 6. Then you can write it out. It is harder for kids to subtract larger quantities, like 7 - 6. So it is very important to get them comfortable with the idea of fact families, and also with missing number problems. For 7 - 6, it is easier to "count up," and say, "Start with 6 how many more to make 7?" Turn that subtraction problem into an addition problem. The number line is great to use, but it is definitely trickier to subtract on a number line than to add with one. Sometimes kids have a hard time knowing where to begin counting backwards when subtracting on a number line. Do you ever use Cuisenaire rods with a centimeter ruler? That way the students can see and feel the "worth" of a number. You can explore the four facts in a fact family easily that way. They can also add and subtract with Cuisenaire rods without using a ruler/number line. Using the same problem as above, you could set out a black (seven) rod, and right above it you could set out a dark green (six) rod. Ask the student what rod is needed next to the green rod so that the two together will equal the length of the black rod. The student should have no trouble determining that it is a white (one) rod. Now while those rods are set up, you can look at the four facts: 6 + 1 = 7. If you reverse them: 1 + 6 = 7. If you take away the white: 7 - 1 = 6. Take away the green instead: 7 - 6 = 1. One confusing thing about subtraction in word problems is that sometimes we are "taking away," but sometimes we are comparing to see which group is larger. So while you are continuing to do lots of manipulative activities to build these concepts, you should keep in mind that you need to model both types of subtraction. With the Cuisenaire rods example above, you could model it both ways. When you remove the white (one) rod, you are taking one away. But you are also comparing the remaining two rods and seeing that the black (seven) rod is one unit larger than the green (six) rod. It's good that you've been doing a lot of work with manipulatives, and I'm sure you'll keep doing a lot more. In the early grades, the more work with manipulatives the better, even for children who "get it." I have observed some young children having a difficult time solving simple problems with manipulatives such as small blocks. I think in some cases they are still developing their fine motor skills, and it is taking most of their concentration just to pick up and grasp the objects in their hand while they are "taking away," so they aren't really focusing on the specific math problem at hand. Maybe you could use the lid of an egg carton for setting up the initial problem (7 objects), and then use another container like a paper cup for holding the objects being removed. That way if a child is solving 7 - 6, he/she can pick up one at a time from the 7 in the egg carton lid, and place one object at a time in the cup while counting out six of them, until one is left in the lid. (When they are in the cup they are harder to see and seem to be more "removed" than if they are just pushed aside.) You could call the cup the "trash can" or something so it seems more as if these objects are being discarded from the problem. > My third and fourth grade students are having so much > trouble with basic subtraction facts (with answers 20 or > less). I am trying fact families, flashcards, Saxon > methods . I am at a loss. Please share great ideas for > helping students learn subtraction facts. It's hindering > them now that they are learning large number subtraction with > borrowing. Thank you. aka Use a Mad Minute test everyday. I teach the second grade and have given a Mad Minute test everday this year. Every single one of my second graders knows those facts and can rapidly recall them! Molly, on teachers.net primary elementary board Let's see if I can remember this great strategy that I used to teach 3rd graders. Okay, if the problem was 15 - 8: We'd draw a ladder with 3 rungs. The bottom rung would be labeled 8, the middle 10 and the top 15. They knew it was 2 steps from 8 to 10, and 5 steps from 10 to 15, so 2 + 5 = 7. One more: 12 - 7 bottom step 7, middle step 10, top step 12 3 + 2 = 5 The key was building off of the steps from 10 which they knew. It really worked and easily transferred to a mental strategy. Good luck, > Anyone know of a site where I can find some printable math > practice worksheets for my second graders? I'd like to > give them quick practice every day so hopefully they will > learn their math facts. I agree that daily quick practice and learning the facts cold is important. When I used to teach second grade math, it took some of my students months to be solid with the basic addition and subtraction facts up to 9 + 9. But we don't need drill worksheets to accomplish fact-learning. At second grade level, a child who does not know the facts cold can still benefit from lots of hands-on work with manipulatives. I used to find it very helpful to use Cuisenaire rods with number tracks or centimeter rulers. (Number tracks hold the rods in a "train." For the problem 8 + 6, for instance, a child would place a rod of the value 8 (8 cm long) in the track or alongside the centimeter ruler. Then the child would place a six rod end-to-end with it and see that the combined length was 14 cm. I would have the child take it one step further: above the 6 cm rod, the child would place two rods: one to reach 10 (the 2) and one for the remainder (4). Now the child had a visual picture that 8 + 6 = 8 + 2 + 4. Why? Because at the same time, the students were developing the concept of using 10 as an "anchor" in solving mental math problems and were also learning to visualize a mental number line. When they became adept at this, it was much easier for them to add two-digit numbers in their heads, later on. For instance: 28 + 36 Along with the usual pencil and paper method involving regrouping, I would also teach them to mentally add the 20 + the 30 to reach 50 and then to add on the 8, to reach 58, and then the 6, to reach 64. This final step was easy if they had had lots of hands-on experience with the sort of problem I described above, breaking the problem into pieces to reach a "10." And this same skill made it much easier for them to begin learning multiplication tables near the end of second grade, or for my third-grade students in other years. For instance, skip-counting by 6's, we'd first do it on a number line (pausing at each 10 along the way): and then we would see that the same pattern repeated again for the sixes from 30 to 60. As a parent, when my oldest child was in first grade and was expected to memorize the addition and subtraction facts, his most meaningful work was done in the bathtub! I used the same strategy with my second and third grade students to help them learn and UNDERSTAND the facts, but of course without the bathtub setting. But if any parents are reading this, I still think nothing beats bathtub math! It worked like this: As I sat beside the tub while my son was soaking and playing, which was a nice, relaxing time (either before or after the getting-clean part), I would ask him fact problems. As soon as I'd say the question, I would start lightly slapping my leg, about one slap per second. He would try to answer within ten slaps. If he couldn't, then he could take as much time as he needed (I'd stop slapping my leg of course), but along with the answer he would need to verbalize an explanation of how he arrived at the answer. No fingers allowed. There were three basic strategies which we found useful: 1) using 10 as an anchor, as I've described above 2) building on a known fact (often a double) 3) starting with 10 + a number and adjusting For strategy #2: if I said "7 + 8," his answer might be, "Well, I knew that 7 + 7 is 14 (because the doubles are easy to learn), so I added one more, because 8 is one more than 7. So it is 15." Similarly, for 7 + 6 he might use 7 + 7 as his known fact and then subtract 1 to get 13, because 6 is one less than 7. For strategy #3: if I said "9 + 8," he might say, "10 + 8 would be 18, so 9 + 8 is one less, 17." Gradually we would shorten the number of slaps, working our way down to 3 seconds, as more and more facts became familiar friends. After we did this each evening for a week, he could answer any fact within three seconds. Either he had simply learned them from the repetition, or he had become so adept at the mental strategies that he could implement them within moments. And these same strategies served him (and my students) well in a variety of ways throughout the years. As a teacher, I sent home information sheets to the parents describing some of these strategies for learning the facts and asked them to reinforce them at home. (I may have even recommended the bathtub!) In the classroom, for your quick fact drill every day (after plenty of hands-on work), you could simply pair students and have them take turns asking each other math facts and answering within an agreed-upon seconds (or verbalizing a strategy). You could give each pair of students one fact sheet and have them take turns asking random facts from the sheet. You could use these same sheets all year, because they never need to be written on. But they really don't even need that. I am teaching an undergraduate course, and several of the participants want those addition (and multiplication) rhymes to help their students remember those trouble-some facts. I have the following, but would be interested in hearing others. Doubles Addition Rhymes 1 Snickers bar +1 Milky Way ___________ makes 2-th decay 2 socks +2 shoes _____________ "4" you to choose 3 little pigs +3 little wigs ____________ for the "6-sly" wolf 4 dinner +4 lunch _____________ 1 "8" a bunch 5 on the left +5 on the right _____________ "10-nd" to fill my gloves just right 6 for myself +6 for my cousin __________ makes 12, that's a dozen 7 little cups +7 little plates ___________ "14-y" weeny dwarfs (for teen-y) I 8 and I + 8 _________ til I was "16" (sick-steen) 9 good guys +9 bad guys ____________ on the "A team" (18) 10-t (tent) +10-t (tent) _________ makes 20 (1-T) KathyB/1st/IA, on teachers.net math board - - - - - Doubles Rhymes Invented by Ian, grade 2 1 + 1 = 2 Zebras belong in the zoo. 2 + 2 = 4 I see skeletons at the door. 3 + 3 = 6 Mother has the heavy bricks. 4 + 4 = 8 It is time to close the gate. 5 + 5 = 10 Dad has a little red hen. 6 + 6 = 12 Who is going to ring the bell? 7 + 7 = 14 Give the crown to the queen. 8 + 8 = 16 Will you eat a big green bean? 9 + 9 = 18 Mr. and Mrs. Duck are dating. 10 + 10 = 20 I like peaches, like 'em plenty. We made this into a book and the kids memorized the poems. It worked! The kids know their doubles by heart. > I am being observed on Wednesday and would like to know if > anyone has an interesting lesson for fact families for > third grade. My principal likes lessons that are done "differently." > Michele (I think I might have got this from this board last year. ) You make a house pattern, then write one fact in each of 4 windows. the 3 numbers in the "family" are written on the roof (a triangle shape). You could be simple or more complicated. When I did it the kids used small pre-cut pieces of fabric for "curtains." They were glued at the top and could be lifted to view the facts. But this may be too many steps for an observation. depends on your kids. LL/NYC, on teachers.net primary elementary board Fact Family Album and Card Game I just did fact families and this is a fun language connection. My students made a fact family album. I gave each student a pair of numbers. They found the third member of their fact family by adding the two together. I then gave the class this cloze passage: We are a fact family. There are three members in our family. Their names are 5, 4, and 9. We can do four things together -- 5 + 4 = 9 4 + 5 = 9 9 - 5 = 4 9 - 5 = 4 They then illustrated their fact family. I did this on the writing paper that has the space at the top for illustrating. I am going to make a class book out of it. Another idea is to use a deck of cards, taking out the face cards. Give each student 2 cards. They figure out what the third member of their fact family is and then pass the cards or lay them down. You can make it competitive by having them keep track of how many fact families they complete. Jessica/3/MO, on teachers.net primary elementary board Our math book has one lesson on "fact families" but I think it's very important to spend more time on this concept. We do fact families for a few minutes each day for about two weeks. The students write them daily in their math notebooks and then we do activities. We have made the houses above and sometimes use spinners. Each student has a spinner and spins two times. He adds the two numbers together for the family and writes the other three sentences. Ex: Spin 3 and 4. Write: 3+4=7 4+3=7 7-3=4 7-4=3 If they get the same two numbers then one of the numbers has to be written in colored pencil so they understand it is a different number. They love spinners! Kids hold math symbol sheets and arrange themselves We just did a lesson last week where we thought of 3 or 4 fact families. Then we put each one on 9 by 12 construction paper, one number on each piece. We used a different color for each problem. We also had a paper with a plus sign, equal sign, and subtraction sign. I'd say a problem---ex. 9 + 4. The kids with the right numbers had to come up front and arrange themselves in the right order to make the fact family--switching their order each time and also switching to the subtraction sign when necessary. They loved it and it really helped. Good luck. anonymous, on teachers.net primary elementary board > HELP - I need some NEW and INTERESTING ideas for teaching fact families. > Dan Have you tried giving each child a couple of dominoes and have them write the fact families they depict? They can turn the dominoes around to see how the addends shift places. Grace/IL, on teachers.net primary elementary board Upstairs and Downstairs Windows I have the kids cut out a house shape (construction paper) and they write the two addition facts in upstairs windows, two subtraction facts in the bottom windows. Also www.mathstories.com [Note: mathstories.com now requires a paid subscription.] These are helpful. Good luck! Kim S, on teachers.net primary elementary board aaamath.com is a good site for elementary children. You can have levels from K to upper elementary. They have many different skills to practice. Hope this helps!! Erin, on teachers.net primary elementary board > Has anyone had experience using Touch Math with first > graders for addition and subtraction? What do you think? I > would appreciate any feedback. Thanks! > Virginia I have used it for years and really like it. I also do other things (fact memorization, etc. because of district requirements) but I started using touch math years ago when I relocated to another state and stumbled on a teacher's edition in my new classroom. The subtraction can get a little tricky at first, but they usually catch on fairly quickly. I also use a lot of manipulatives in math, but this program helps fill in the gaps for students who have difficulty memorizing and for those who need something more concrete. By the way. teach both forms of the number 4 (closed top as here and open top) so they will be able to use the points with any form of typing/writing. Do the same with the 9/6. They need to be able to visualize the points on various types of number formation/writing. ww, on teachers.net math board Touch Math Hurts Students' Future Chances in Math I teach upper level math in school ( 7 - 12 math) and I have come to realize that touch math seriously hurts the students' future ability in mathematics. Students must memorize basic math facts to the point of automaticity. I discovered that the elemantary teachers were using this method because memorization has fallen out of favor in trendy educational circles. anonymous, on teachers.net math board Re: Touch Math Hurts Students. -- not if used properly If used properly, Touch Math is a supplemental program to be used along with traditional methods of teaching -- not as a replacement. I found that my students DID learn the required facts as we used Touch Math. I now teach at-risk students. I have one child in third grade who has been struggling to learn math. She came to me a week ago with an attitude of hatred toward math. She simply didn't get it. And they were working on addition of two and three digit numbers. I spent a half hour showing her Touch Math. We worked some problems together that day and she completed a math worksheet on her own. She took it back to the teacher and made a 100 on it. She was ecstatic. She has come to me for 30 minutes each day. In four days she had done a complete turn around. She is now excited about math and actually looks forward to doing it. Today they began subtraction of two and three digit numbers. I showed her backward counting with Touch Math. Again the elation as she began to complete the assignment independently. Will she learn the basic math facts? Yes, because her teacher and I are both strong believers in memorization of math facts. I challenge the poster who is negative about Touch Math to prove that it is the Touch Math that is destroying students -- or is it possibly that the teachers he/she is aware of are using it improperly? Sharyl, on teachers.net math board Touch Math hurts students' chances by not learning basic facts What is the goal? If the goal is to learn basic math facts to automaticity, then do that. Touch math only teaches them a process that will allow them to get the answer without memorizing the basic math facts to automaticity. We must teach what we want the students to learn. The first group of touch math kids are starting addition and subtraction of mixed numbers, division of decimals, and algebra this year. They are so slow and are so poorly prepared to deal with basic facts that they are NOT getting it. Those who know the basic facts effortlessly acquire the new concepts. The touch math kids are tip tip tipping their way into an endless spiral of frustration. l anonymous, on teachers.net math board I would check with your lower grade teachers then. When I utilized Touch Math as a SUPPLEMENTAL program in Oklahoma, our students also learned the basic facts. As they progressed upward on into junior high and high school, they had no problem with it. And I will go one step further, even though I can see that you have your mind already made up against Touch Math. When I was in 3rd/4th grade (a split class in Marland, Oklahoma) back in the late 1950's, our teacher, Mary Nell Lewis, taught us the exact same system as Touch Math. (Too bad she didn't patent it!) Yet isn't it amazing that even knowing a system like Touch Math, we were all able to still learn the basic math facts and be very successful in all areas of math, even as we progressed through school into higher level math. Do you have a problem with allowing students to count on their fingers until they learn the basic math facts? I do. I would prefer they utilize Touch Math rather than counting on fingers. As I stated earlier, Touch Math should be used as a SUPPLEMENTAL program. It doesn't and shouldn't replace learning basic math facts. My question to you is this -- what do you propose students do until they learn the basic math facts? Perhaps you would prefer that they not attempt to work any math problems until they've learned the facts? Sharyl, on teachers.net math board Touch Math helped my daughter First I'd like to say that I'm not a teacher. My daughter in second grade was having a hard time adding on. She started using the number line and this helped. I soon realized that she was having a difficult time doing addition without this visual aid. I learned about touch points from a web site. In one week I couldn't believe the difference in her ability to add. What I like about touch points is you don't need any supplemental object to use it. The numbers are always on your paper. I would rather have my daughter use touch points than use her fingers. I've also found she is starting to memorize facts. She knows all of her double facts and the addition facts are coming. I've started using touch points with money. A nickel has 1 touch point. A dime has two touch points and a quarter 5 touch points. Each touch point the child counts by five, for those coins. A penny has one. She can count any amount of money now. She is starting to take two quarters and add from.50. She is making progress.

Why do teachers alway think there is only one way to teach? When a child learns to ride a bike we give them training wheels. Eventually, we take away the training wheels and they ride without them. A child with a broken leg needs a crutch. When the leg heals we remove the crutch. I believe the same is true with teaching math. If a child needs support-GIVE THEM SUPPORT TO HELP THEM BE SUCCESSFUL

Val G. Just a mom, on teachers.net math board

Thank you for your input. I agree wholeheartedly.

I feel like THAT is the first step. If we can get them LIKING Math we can help them LEARN Math.

As I said earlier, if Touch Math is used appropriately, it is a SUPPLEMENTAL method which supports traditional teaching. Those math facts should still be taught/learned, but in the mean time, Touch Math does away with counting on fingers. I, for one, despise the counting on fingers.

Sharyl, on teachers.net math board

I was rather dissapointed by the comment that touch math does nothing to help students learn and memorize their facts -- It helped me tremendously when I had excessive difficulty with math as a young student and HAVE seen it work in other instances as well as a tutor myself! -- Once the actual "touch" part is understood and mastered, students can simply visualize the points on the numbers and figure the answers with automaticity. While some students may struggle to get past counting the spots on the numbers by a certain stage, I have seen this strategy work not only for myself but with others as well.

Jennifer P., message to Math Cats

Touch Math hurts my daughter - she can't "undo" it

I was "led" to this site looking for information on Touch Math -- and how to "undo" it. My daughter learned math by Touch math in 2nd and 3rd (and 4th) grades and of course it helped her tremendously then. But now she is in 8th grade -- and still needs to do her addition and subtraction this way. You cannot be very successful at 8th grade math using touch math as your basis. We are at a loss trying to figure out how to "undo" this Touch math and help her to move on.

I see all of the postings regarding this subject are quite old. but thought someone might still tune in on the subject.

Addition and Subtraction with Decimal Numbers

Addition and subtraction with decimals are just the same as addition and subtraction with the whole number. The most important thing to remember is to line up the decimal points so that when you stack them, they are directly on top of each other. Doing this stacks the place values so that the ones are on top of the ones, the tens on the tens, the tenths on the tenths, and so forth. If there are any times when there aren’t as many digits after the decimal place in one number as there are in another, remember that you can fill in any places to the right of the decimal with a 0. In numbers, this looks like 3.15 = 3.150 = 3.150000000 and so on.

Steps for Addition (Combining Two Positive or Two Negative Numbers)

1. Stack the numbers according to place value. (This will line up the decimals.)
2. Add the numbers in the columns starting with the column on the right.
3. Carry numbers into the next place value to the left as needed.
4. Repeat this pattern for each remaining column.
5. Bring the decimal down.
6. Check sign. (If both numbers were positive, the answer is positive. If both numbers were negative the answer is negative.)

Steps for Subtraction (Combining a Positive and Negative Number)

1. Biggest on top.
2. Stack in columns according to place value. (This will line up the decimals.)
3. Regroup when needed.
4. Subtract in columns by place value starting on the right and going left.
5. Bring the decimal down.
6. Strongest wins. (If the bigger number was negative then the answer is negative. If the bigger number was positive then the answer is positive.)

Evaluate the following expression:

Solutions

Using place values, we will add 2.4 and 8.8. Line up the numbers in their corresponding places.

Next, add the numbers in the tenth column. Since 4 + 8 = 12, place the 2 below the tenths place and the 1 on the ones place.

Finally, add the numbers in the ones place. Include the 1 carried over from the tenths place.

Be sure to include the decimal point between the tenths place and the ones place.

Using place values, we will add 8.12 and 9.81. Line up the numbers in their corresponding place values.

Next, add the numbers in the hundredths column. Since 2 + 1 = 3, place the 3 below the hundredths place.

Next, add the numbers in the tenths column. Since 1 + 8 = 9, place the 9 below the tenths place.

Finally, add the numbers in the ones column. Since 8 + 9 = 17, we put a 7 in the ones place and the 1 in the tens place.