What can I do to help my child with fact and computational fluency at home

What can I do to help my child with fact and computational fluency at home?

The state of Ohio requires certain levels of fluency at each grade level.

Fluency is defined as efficiency, accuracy, and flexibility.

Although repetitive practice can be helpful once students know facts and computational strategies, repetitive practice will not help students to learn facts and computation in the first place, at least not if our goal is for students to remember them long-term and apply them appropriately in a variety of situations.

So, it is extremely important to help students learn facts and computation in meaningful ways before repetitive practice is required. The information below will provide you with ways to work with children at each grade level.

**If you would like a paper copy of anything on this page, you can simply copy and paste the text that you want into a Word document; then print it.

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Kindergarten

No formal fact or computational knowledge is required at this level. However, some very beneficial activities with your child are:

Have your child count objects, and ask your child to count the same objects after they are moved around to see that the total has not changed. Then have your child write the number down and say it again so that connections are made among the concrete, word, and symbolic representations of the number.
Give your child a number from 0-19, and ask him/her to tell you the next number. (You say 7; child says 8.) Then have your child write both numbers and say them again. The more challenging variation of this is to ask your child to give you the number before (1 less than) the number you have said.
Using a number cube with up to 6 dots on a side, have your child tell you how many dots are there on a side. Then have your child write the number down and say it again. The more that your child can quickly recognize the number of dots without counting (after doing this many times), the more that he/she will be able to use that mental image of the number in thinking about and solving problems. Very beneficial! If your child masters the number cube, using dominoes is a good way to move this activity up a level.
Give your child two numbers between 0 and 10, and have him/her tell you which is larger. At first, it may be helpful for your child to count objects for each number to be sure that he/she understands why the larger one is larger.
Put out a number of objects (between 0 and 10), and ask your child to add 1 or 2 to the set and tell you how many are there now. Then have your child write the original total and the new total.

Grade 1

In first grade, we start to develop strategies for learning addition and subtraction facts with students. In our fluency plan, we work on several specific strategies that are also in the Ohio standards. (See these strategies below.) The state does not require any memorized facts by the end of grade 1.

Add zero: Have your child put out between 1 and 10 objects and add nothing to them. Have him/her say the original total and the new total. Then, have your child write a number sentence that shows what the model shows (for example: 7 + 0 = 7).
Add to zero: Have your child start with nothing on the table; then have him/her add between 1 and 10 objects. Have him/her say the original total and the new total. Then, have your child write a number sentence that shows what the model shows (for example: 0 + 5 = 5).
Turn-around facts (actually the commutative property): Once your child starts learning addition facts, it is helpful to discuss with him/her what happens when two numbers are added in either order (for example, 6 + 0 or 0 + 6). Have your child write examples such as 0 + 3 = 3 + 0. This should not be done, however, without modeling these facts with concrete items first and talking about them.
One more: Have your child put out between 1 and 10 objects and add one to them. Have him/her say the original total and the new total. Then, have your child write a number sentence that shows what the model shows (for example: 4 + 1 = 5).
Two more: Have your child put out between 1 and 10 objects and add two to them. Have him/her say the original total and the new total. Then, have your child write a number sentence that shows what the model shows (for example: 6 + 2 = 8).
Doubles (addition): Doubles facts are often easy to remember for students, and they help to build a foundation for many other facts. Dominoes and many other real-life items that come in pairs (like juice boxes, eggs, pudding cups, etc.) are great for helping students to picture and learn these facts. Have your students look for or create models of doubles and then write number sentences for them. (for example: 6 + 6 = 12)
Subtract zero: Have your child put out between 1 and 10 objects and take away nothing from them. Have him/her say the original total and the new total. Then, have your child write a number sentence that shows what the model shows (for example: 9 - 0 = 9).
Subtract all: Have your child put out between 1 and 10 objects and take away all of them. Have him/her say the original total and the new total. Then, have your child write a number sentence that shows what the model shows (for example: 4 - 0 = 4).
One less: Have your child put out between 1 and 11 objects and take one away from them. Have him/her say the original total and the new total. Then, have your child write a number sentence that shows what the model shows (for example: 7 - 1 = 6).
Two less: Have your child put out between 2 and 12 objects and take two away from them. Have him/her say the original total and the new total. Then, have your child write a number sentence that shows what the model shows (for example: 11 - 2 = 9).
Math families: It is VERY important that your child develops an understanding of the relationship between addition and subtraction – the idea that subtraction is a reversal of the addition process, and vice versa. So, with any of the facts above, we often ask students to write math families that include two addition and two subtraction facts with the same three values. It will be most helpful for students to have concrete objects in front of them to model these facts as they write them. For example, if the fact being explored is 5 + 2 = 7, the math family would include 2 + 5 = 7, 7 – 5 = 2, and 7 – 2 = 5. It is also important that your child knows that these can be written as 7 = 5 + 2, 7 = 2 + 5, 2 = 7 – 5, and 5 = 7 – 2. However, it is often difficult at first for students to understand this; what often helps is to help them think of the equal sign meaning is the same as or balances. (7 is the same as 5 + 2, or 7 balances 5 + 2.)
Doubles (subtraction): Again, have your child model/look for doubles facts (see above). However, this time, ask your child to think of subtraction facts that relate the whole set to the value that is doubled and write a subtraction fact. (Math families – above – will also help with this.) For example, if a student models 4 + 4 = 8, it is easy to see that 8 – 4 = 4. Have your child say this and write the number sentence.
Make 10: One of the most important foundational concepts for students at this age is to understand what numbers can be combined to make 10. This helps to develop place value understanding and also assists in mental and paper/pencil computation later on. There are many ways to help students explore these combinations: One way is to use 2 rows of 5 objects, where some of the objects are one color and some another color. (Ex: 6 blue and 4 green, so they would say and write 6 + 4 = 10, 4 + 6 = 10, 10 – 6 = 4, and 10 – 4 = 6.) Similarly, students can use their 10 fingers – hold some up and put some down; then say and write the matching number facts. When students have had some of these experiences, you can practice with them with card games like Tens Go Fish, where you only use cards that have values of 10 or less, and the goal is to collect pairs of 10. (If a player had a 7, she could ask the next player for a 3.) Even in the car, you can say numbers and have your child tell you what numbers go with them to make 10 (ex: you say 2; child says 8).

Click here for a Microsoft Word document that lists many fun ways to practice facts once students have learned them. See below for some websites that provide fluency practice.

Other helpful activities for grade 1, any time, any place:

How much is needed to get the next ten? Give your child a number between 0 and 100, and ask how much is needed to get to the next (multiple of) ten. For instance, if you said 7, your child would say 3. If you said 82, your child would say 8. This is a very helpful element of developing mental computation skills. Most children will have to start with relatively low numbers. An appropriate extension for grade 1 would be to ask how much is needed to get to 100.
What is 10 more? 10 less? 20 more? 30 less? Give your child a number between 0 and 100, and ask what number would be 10 more than that number (or 10 less, or 20 less, or 50 more, or any multiple of 10 more or less). This is a great place value activity and also helpful in developing mental computation skills. Most children will have to start with 10 more and 10 less and later work up to higher values. An appropriate extension for grade 1 would be to ask about (for instance) 15 more, 21 less, etc. This activity can also be done in the context of money (what is 10 cents more?).

Grade 2

By the end of second grade, the state of Ohio expects that students are fluent with the addition and subtraction facts to 10 + 10 and 20 – 10. Along with the strategies listed above for first grade, we use the strategies below to help students reach this goal.

Near doubles: A near doubles fact is a fact that is one away from a doubles fact. For instance, 4 + 5 = 9 (along with 5 + 4 = 9, 9 – 5 = 4, and 9 – 4 = 5.) It is VERY helpful for students to connect these directly to the doubles facts instead of learning them separately. One way to help your child explore these facts is to have him/her model a doubles fact, such as 5 + 5 = 10, with concrete objects; then, either add 1 to one set of 5 or take 1 away. Then, your child should say out loud what the new model shows and write the addition and subtraction facts that relate to the model (like the four above).
Two apart facts: We can either think of these facts as two away from a double, or one up and one down from a double. An example of a two apart fact is 5 + 7 = 12 (along with 7 + 5 = 12, 12 – 5 = 7, and 12 – 7 = 5). For this example, we could start by asking a child to model 5 + 5, add 2 to one set of 5; then say out loud and write the four related addition and subtraction facts. OR, we could ask the child to model 7 + 7, take 2 away from one set of 7; then say and write the four facts. OR, we could ask the child to model 6 + 6, take 1 from one set of 6 and put it with the other set; then say and write the four facts. For the best chance of making solid connections in the mind of the child, it is best to do ALL of these at various times.
Using 10 as a landmark: This strategy helps to learn most of the facts that might be most challenging for students to remember. (The Make 10 strategy above in first grade is an important precursor to this.) Concrete objects and the number line are helpful models for this strategy. An example fact in this set could be 5 + 8 = 13. There are many ways to think about this fact, but using a concrete model with this strategy could involve setting up two 10-frames (each with two rows of 5 empty spaces). Have your child fill in 5 of the spaces in one frame with objects; then remind him/her that 8 need to be added. Have your child fill the frame the rest of the way – using 5 objects – then place the final 3 in the second frame. This is a very visual way to help the student use 10 as a landmark to find the sum. Again, the child should then say and write 5 + 8 = 13. (8 + 5 can also be modeled in the same way.) It is easy to model the reverse subtraction problem with this model also – in the example 13 - 8, start with 13, take away 3 to get to 10, then take away 5 more to get 5. Then write and say the fact 13 – 8 = 5. (13 – 5 can also be modeled in the same way.) Going through this process may seem time-consuming, but it will help your child to firmly establish 10 as an important value in thinking about many types of arithmetic problems.

Click here for a Microsoft Word document that lists many fun ways to practice facts once students have learned them. See below for some websites that provide fluency practice.

Also, by the end of second grade, the state of Ohio expects that students are learning strategies for double-digit addition and subtraction. Fluency is not expected until the end of grade 3. Both through student-generated strategies and teacher facilitation, we encourage the following types of strategies (these are examples and should not be considered a complete list). Many of these strategies are easy to use mentally. The names/descriptions of each strategy are provided for adult reference only.

For addition:

To add 45 + 39: 44 + 40 = 84 (this is called compensation since 1 was added to 39 and taken from 45)
To add 38 + 71: 40 + 69 = 109 (another example of compensation since 2 was added to 38 and taken from 71)
Add tens; then add ones: 57 + 38 = 50 + 30 + 7 + 8 = 80 + 15 = 95 (students often do this naturally before any instruction)
Add tens; then add ones from first addend; then add ones from second addend: 94 + 38 = 90 + 30 + 4 + 8 = 120 + 4 + 8 = 124 + 8 = 132)

For subtraction:

To subtract 83 – 19: 84 – 20 = 64 (this is also compensation since 1 was added to both 83 and 19 – the difference will still be the same)
To subtract 50 – 22 = 48 – 20 = 28 (this is also compensation since 2 was subtracted from both 50 and 22 – the difference will still be the same)
Subtract tens of smaller number, then subtract ones – to find 52 – 37: 52 – 30 = 22; 22 – 7 = 15
Count up to subtract – to find 134 – 58: 58 to 60 is 2; 60 to 130 is 70; 130 to 134 is 4; the total is 76) – this strategy really emphasize the difference or distance between the two numbers (visualize this on a number line)

Other helpful activities for grade 2, any time, any place:

How much is needed to get to 100? Give your child a number between 0 and 100, and ask how much is needed to get to 100. For instance, if you said 35, your child would say 65. If you said 71, your child would say 29. This is a very helpful element of developing mental computation skills. Most children will have to start with multiples of 5 and 10, but work in school on the 100 chart will help them start with other numbers also. An appropriate extension for grade 2 would be to start at numbers between 100 and 200 and ask how much is needed to get to 200.
What is 10 more? 10 less? 20 more? 30 less? Give your child a number between 0 and 100, and ask what number would be 10 more than that number (or 10 less, or 20 less, or 50 more, or any multiple of 10 more or less). This is a great place value activity and also helpful in developing mental computation skills. Most children will have to start with 10 more and 10 less and later work up to higher values. An appropriate extension for grade 2 would be to start at numbers between 100 and 200 and ask the same types of questions. This activity can also be done in the context of money (what is 10 cents more?).

Grade 3

Grade 3 is a challenging year for students in terms of the Ohio standards. During this year, students are expected to learn the following:

∑ Fluency in multiplication and division facts (to 10 x 10 and 100 divided by 10)

∑ Fluency in two- and three-digit addition and subtraction

∑ Fluency in multiplying and dividing double- and triple-digit numbers by single digit numbers

And these are just three of the twelve indicators in the Number standard – along with indicators in Measurement, Geometry, Algebra, and Data!

For multiplication and division facts (listed in the order that generally makes sense for most students and that we use as part of the fluency plan):

It is important that students model and describe these facts concretely before the notation is introduced, and when notation is introduced, students should be able to verbalize that we are finding groups of a given value (multiplication) or how many groups of one value are in another (division). The symbol / is used below for the division sign because not all web browsers can read the division sign as a symbol.

Doubles: These facts relate to the doubles addition facts (see grade 1 above) because we are multiplying by 2. For instance, 7 x 2 = 14. Have your child put out 7 objects and then 7 more (in rows), say 2 groups of 7 is 14, and write 2 x 7 = 14. Then, have your child model 7 groups of 2 (in rows), say 7 groups of 2 is 14, and write 7 x 2 = 14. Ask your child what is the same and different about these two facts – we are now looking at the commutative property since the values can be multiplied in either order (as in addition). At this point, it is a good idea to bring in the two related division facts as well. You can do this by asking how many groups of 2 are in 14 (child should say 7) and then having your child write 14 / 2 = 7. Similarly, how many groups of 7 are in 14? (2) The child should say this and write 14 / 7 = 2.
Tens: Tens facts tend to be easy for students to remember, but it is important that they learn them with understanding. For instance, your child can use a 10 by 10 grid to model (shade) 3 groups of 10 and find that this is 30; then say this and write 3 x 10 = 30. Doing this with several other numbers will make the point that we can simply append a 0 to the original value to find the product of 10 and that value. (It is not a great idea to say add a 0 because we are not, in fact, adding 0 to the number, and though this makes sense to an adult, it may not make sense to a child who is just learning this idea.) Then, ask your child what 10 groups of 3 would equal (using the grid to check if needed), and have your child write 10 x 3 = 30. Next, ask how many groups of 10 are in 30, and have your child say this and write 30 / 10 = 3. Similarly, how many groups of 3 are in 30? (10) The child should say this and write 30 / 3 = 10.
Zeros: Learning the zeros multiplication facts is not difficult, especially when a child can think about them concretely. How many are in 0 groups of 8? 0 groups of 5? 9 groups of 0? 2 groups of 0? For each, have your child say this statement, the product (0), and write the fact, such as 0 x 8 = 0 or 9 x 0 = 0. Be aware that children do tend to confuse these, when not thinking, with the zeros addition facts. For division, we have to be cautious. We can ask how many groups of 8 are in 0, or how many groups of 4 are in 0, etc. These would be written as 0 / 8 = 0 or 0 / 4 = 0. However, we cannot divide BY 0! For example, 6 / 0 = ? This would be like asking how many groups of 0 are in 6; there is no reasonable answer. Another way to look at this: if there were an answer (a quotient) in this problem, we could work backwards and say that that answer times 0 would equal 6. Impossible! So, do not present your child with questions like 5 / 0 = __.
Ones: Again, usually these are not difficult for students, but sometimes they will confuse them, when not thinking, with the one more addition facts. Have your child model 4 groups of 1, for example, say that 4 groups of 1 is 4, and write 4 x 1 = 4. Then have your child model 1 group of 4, and say and write the fact 1 x 4 = 4. For division, ask how many groups of 1 are in 4, and have your child say this and write 4 / 1 = 4. Then, ask how many groups of 4 are in 4, and have your child say this and write 4 / 4 = 1.
Fives: The fives facts should be related to the tens facts since each product of 5 is half a product of 10 (ex: 6 x 10 = 60; 6 x 5 = 30). A way to have your child model these is to use 10-frames (2 rows of 5 empty spaces in each frame) and fill rows of 5. 6 rows of 5 would contain 30 objects. Then, you can ask how many are in 5 groups of 6 (child may need to model this or may not), and the child can say 5 groups of 6 are 30 and write 5 x 6 = 30. Ask how many groups of 5 are in 30, and the child can model this and then say and write 30 / 5 = 6. Finally, ask how many groups of 6 are in 30, and the child can model this and then say and write 30 / 6 = 5.
Nines: The nines facts should also be related to the tens facts since each product of 9 is a product of 10 minus one set (ex: 4 x 10 = 40; 4 x 9 = 36). A way to have your child model this is to use 10-frames (2 rows of 5 empty spaces in each frame) and fill each frame except for one space. 4 sets of 9 would contain 36 objects (40 with 4 missing). Your child should say that 4 groups of 10 is 40, so 4 groups of 9 is 36. Then, you can ask how many are in 9 groups of 4 (child may need to model this or may not), and the child can say 9 groups of 4 are 36 and write 9 x 4 = 36. Ask how many groups of 9 are in 36, and the child can model this and then say and write 36 / 9 = 4. Finally, ask how many groups of 4 are in 36, and the child can model this and then say and write 36 / 4 = 9.
Squares: The squares facts are those where a value is multiplied by itself and where the model is literally a square. For instance, 5 groups of 5 is 25, or 5 x 5 = 25. If a student models this with 5 rows of 5 objects (especially squares!), he/she will create a figure (an array) that is essentially square. Have your child model these and also describe and write the related division facts. In this case, ask how many groups of 5 are in 25, and have the child say this (and the quotient – the answer) and write 25 / 5 = 5.
Doubles and one more set (threes): The threes facts should be related to the doubles facts since each product of 3 is a product of 2 plus one more set (ex: 8 x 2 = 16; 8 x 3 = 24). A way to have your child model this example is to lay out 8 rows of 2; then add a third object to each row. Then, your child should say that 8 groups of 2 is 16, so 8 groups of 3 is 24 (16 + 8), and he/she should write 8 x 3 = 24. Then, you can ask how many are in 3 groups of 8 (child may need to model this or may not), and the child can say 3 groups of 8 are 24 and write 3 x 8 = 24. Ask how many groups of 3 are in 24, and the child can model this and then say and write 24 / 3 = 8. Finally, ask how many groups of 8 are in 24, and the child can model this and then say and write 24 / 8 = 3.
Which are Left? Facts: The only multiplication facts that are not covered by the strategies listed above are 4 x 6, 4 x 7, 4 x 8, 6 x 7, 6 x 8, and 7 x 8 (and the related facts such as 6 x 4, 7 x 4, etc.). This does not mean that we should simply ask students to memorize these facts and the associated division facts! See grade 4 below for ways to work on fours facts and two apart multiplication facts, and any of these facts can be modeled with concrete objects and related to other facts. For example, sevens facts can be related to fives and twos: 6 x 7 is the same as 6 x 5 plus 6 x 2 (this can be understood easily with a model). Another example: 7 x 8 is the same as 7 x 7 plus one more 7. Always encourage your child to make connections to help recall the facts.

Click here for a Microsoft Word document that lists many fun ways to practice facts once students have learned them. See below for some websites that provide fluency practice.

For fluency in two- and three-digit addition and subtraction (grade 3):

Along with the strategies from grade 2 (see above), we encourage the use of the following strategies. The following are examples and not intended to represent every possible strategy. Also see our videos for demonstrations of many of these strategies. The most important goal is that students should be able to use numbers flexibly and efficiently to compute accurately because one strategy may be very easy to use in one problem and not so easy to use in another.

For addition:

Add hundreds, then tens, then ones – ex: 735 + 276: 700 + 200 = 900; 30 + 70 = 100; 5 + 6 = 11; 900 + 100 + 11 = 1011
Add hundreds of second addend to first addend, then tens, then ones – ex: 258 + 594: 258 + 500 = 758; 758 + 90 = 848; 848 + 4 = 852
Compensation (works very easily in some cases) – ex: 258 + 594: 258 + 600 = 858; 858 – 6 = 852
Standard American algorithm WITH understanding – ex: 367 + 459:

∑ 7 + 9 = 16, record the 6, and add the 1 (10) from 16 to the tens column;

∑ 1 + 6 + 5 (which really is 10 + 60 + 50) is 12 (which really is 120), record the 2 (20), and add the 1 (100) to the hundreds column;

∑ 1 + 3 + 4 (which is really 100 + 300 + 400) is 8 (800); answer is 826

For subtraction:

Subtract hundreds, then tens, then ones; then combine – ex: 326 – 179: 300 – 100 = 200; 20 – 70 = -50; 6 – 9 = -3; 200 – 50 – 3 = 147
Find the distance from one value to the other: ex: 807 – 568: 568 to 600 is 32; 600 to 807 is 207; 207 + 32 = 239
Compensation (works very easily in some cases) – ex: 461 – 285: 461 – 300 = 161; 161 + 15 = 176 (since taking away 300 from 461 was 15 too much)
Standard American algorithm WITH understanding – ex: 812 – 654:

∑ Choose not to take 4 from 2, so regroup one ten; 12 – 4 = 8;

∑ Choose not to take 5 from 0 (which is really 50 from 0 tens), so regroup one hundred; 10 – 5 = 5 (which is really 100 – 50 = 50);

∑ Subtract hundreds; 7 – 6 = 1 (which is really 700 – 600 = 100); answer is 158

For fluency in multiplying and dividing double- and triple-digit numbers by single digit numbers (grade 3):

We encourage the use of the following strategies, most of which rely on the distributive property. The following are examples and not intended to represent every possible strategy. Also see our videos for demonstrations of many of these strategies. The most important goal is that students should be able to use numbers flexibly and efficiently to compute accurately because one strategy may be very easy to use in one problem and not so easy to use in another.

For multiplication (several of these can be modeled using the areas of rectangles; see third strategy below for an example):

Repeated addition (only to be used initially) – ex: 82 x 4 = 82 + 82 + 82 + 82 = 328
Decomposing by tens and ones (also used initially) – ex: 37 x 5 = (10 x 5) + (10 x 5) + (10 x 5) + (7 x 5) = 50 + 50 + 50 + 35 = 185
Decomposing by place value – ex. A: 268 x 7 = (200 x 7) + (60 x 7) + (8 x 7) = 1400 + 420 + 56 = 1876; ex. B: 48 x 9 = (48 x 10) – 48 = 480 – 48 = 432

∑ Sketch a rectangle with sides 268 and 7 units; divide this into three rectangles: 200 by 7, 60 x 7, 8 x 7; find the area of each; add these areas

Other partitions – ex: 527 x 4 = (500 x 4) + (25 x 4) + (2 x 4) = 2000 + 100 + 8 = 2108
Standard American algorithm WITH understanding – ex: 273 x 8:

∑ 3 x 8 = 24; record the 4; regroup the 2 (really 20);

∑ 7 x 8 = 56 (really 70 x 8 = 560); 56 + 2 = 58 (really 560 + 20 = 580); record the 8 (really 80); regroup the 5 (really 500);

∑ 2 x 8 = 16 (really 200 x 8 = 1600); 16 + 5 = 21 (really 1600 + 500 = 2100); answer is 2184

For division (the symbol / is used below for the division sign because not all web browsers can read the division sign as a symbol):

Use multiplication/building up – ex: 213 / 8 (How many groups of 8 are in 213?)

∑ 8 x 20 = 160; 8 x 5 = 40; 160 + 40 = 200;

∑ 8 x 1 = 8; 200 + 8 = 208;

∑ 20 + 5 + 1 = 26; so, 26 groups of 8 with 5 left over

Take out multiples of 10 of the divisor – ex: 152 / 6 (How many groups of 6 are in 152?)

∑ 152 – 60 = 92 (took out 10 groups of 6);

∑ 92 – 60 = 32 (took out 10 more groups of 6);

∑ 32 – 30 = 2 (took out 5 more groups of 6);

∑ 10 + 10 + 5 = 25; so, 25 groups of 6 with 2 left

Modified standard American algorithm – ex: 597 / 8 (How many groups of 8 are in 597?) - *Since this is difficult to display as a graphic on a webpage so that all web browsers can interpret it, you may want to jot down the problem in the standard format and follow along with the steps as described below.

∑ Think: How many times does 8 go into 590? (70) Write 70 above the division sign; subtract 560 from 597 to get 37.

∑ Think: How many times does 8 go into 37? (4) Write + 4 after the 70 above the division sign; subtract 32 from 37 to get 5.

∑ The answer is 74 with 5 left over.

Standard American algorithm WITH understanding – ex: 314 / 7 (How many groups of 7 are in 314?) - *Since this is difficult to display as a graphic on a webpage so that all web browsers can interpret it, you may want to jot down the problem and follow along with the steps as described below.

∑ Think: How many times does 7 go into 310 (since it cannot go into 3 evenly)? 40 – record the 4 above the 1 (in the tens place to indicate 40), and write 280 below 314 (instead of just 28) since 7 x 40 = 280.

∑ Subtract: 314 – 280 = 34.

∑ Think: How many times does 7 go into 34? 4 – record the 4 above the 4 in 314, and write 28 below 34 since 7 x 4 = 28.

∑ Subtract: 34 – 28 = 6. The answer is 44 with 6 left over.

Other helpful activities for grade 3, any time, any place:

How much is needed to get to the next multiple of 100? Give your child a number between 0 and 1000, and ask how much is needed to get to the next multiple of 100. For instance, if you said 342, your child would say 58. If you said 210, your child would say 90. This is a very helpful element of developing mental computation skills. Most children will have to start with multiples of 5 and 10, and some may need to start with numbers less than 100, but playing the game Close to 100 will help them start with other numbers also. An appropriate extension for grade 3 would be to ask how much is needed to get to 1000.
What is 10 more? 10 less? 20 more? 30 less? 100 more? 400 less? Give your child a number between 0 and 1000, and ask what number would be 10 more than that number (or 10 less, or 20 less, or 50 more, or 100 more, or 200 less, or any multiple of 10 or 100 more or less). This is a great place value activity and also helpful in developing mental computation skills. With three-digit numbers, most children will have to start with 100 more and 100 less and later work up to higher values and 10 more/less (it is a bit harder to find 10 more/less when the numbers are in the hundreds since children tend to focus on the hundreds). An appropriate extension for grade 3 would be to ask about (for instance) 150 more, 220 less, etc. This activity can also be done in the context of money (what is 50 cents more?).

Grade 4

Grade 4 is the last year in the Ohio standards that includes whole number computation. During this year, students are expected to learn the following:

∑ Fluency in multi-digit addition and subtraction

∑ Fluency in multiplying and dividing by double-digit numbers and multiples of 10

We also review multiplication and division facts in fourth grade. Along with reviewing some of the strategies developed in grade 3 above, we develop the following strategies in grade 4. (The symbol / is used below for the division sign because not all web browsers can read the division sign as a symbol.)

Squares and one more set: In this set of facts, each product is a square plus one more set of the number being squared (ex: 5 x 5 = 25 which is 5 sets of 5; 5 x 6 = 30 which is 5 sets of 6, or 6 sets of 5). A way to have your child model this example is to lay out 5 rows of 5; then add a sixth object to each row. Then, your child should say that 5 groups of 5 is 25, so 5 groups of 6 is 30 (25 + 5), and he/she should write 5 x 6 = 30. Then, you can ask how many are in 6 groups of 5 (child may need to model this or may not), and the child can say 6 groups of 5 are 30 and write 6 x 5 = 30. Ask how many groups of 5 are in 30, and the child can model this and then say and write 30 / 5 = 6. Finally, ask how many groups of 6 are in 30, and the child can model this and then say and write 30 / 6 = 5.
Doubles doubled (fours): The fours facts should be related to the doubles facts since each product of 4 is twice a product of 2 (ex: 3 x 2 = 6; 3 x 4 = 12). A way to have your child model this example is to lay out 3 rows of 2; then double the size of the model (3 rows of 4). Then, your child should say that 3 groups of 2 is 6, so 3 groups of 4 is 12 (6 x 2), and he/she should write 3 x 4 = 12. Then, you can ask how many are in 4 groups of 3 (child may need to model this or may not), and the child can say 4 groups of 3 are 12 and write 4 x 3 = 12. Ask how many groups of 4 are in 12, and the child can model this and then say and write 12 / 4 = 3. Finally, ask how many groups of 3 are in 12, and the child can model this and then say and write 12 / 3 = 4.
Two apart multiplication facts: The two apart multiplication facts should be related to the squares facts since each two apart fact is the square of the number in between the factors, minus 1 (ex: 7 x 7 = 49; 6 x 8 = 48, or 49 - 1). In this case, 6 x 8 is the two apart fact. A way to have your child model this example is to lay out 7 rows of 7; then take one away from the corner and move the rest in that column down to create an eighth row. Then, your child should say that 7 groups of 7 is 49, so 6 groups of 8 is 48 (49 - 1), and he/she should write 6 x 8 = 48. Then, you can ask how many are in 8 groups of 6 (child may need to model this or may not), and the child can say 8 groups of 6 are 48 and write 8 x 6 = 48. Ask how many groups of 6 are in 48, and the child can model this and then say and write 48 / 6 = 8. Finally, ask how many groups of 8 are in 48, and the child can model this and then say and write 48 / 8 = 6.

Click here for a Microsoft Word document that lists many fun ways to practice facts once students have learned them. See below for some websites that provide fluency practice.

For fluency in multi-digit addition and subtraction (grade 4):

The following are extensions of the strategies from grade 3 (see above). These are examples and not intended to represent every possible strategy. Also see our videos for demonstrations of many of these strategies. The most important goal is that students should be able to use numbers flexibly and efficiently to compute accurately because one strategy may be very easy to use in one problem and not so easy to use in another.

For addition:

Add thousands, then hundreds, then tens, then ones – ex: 2478 + 1356: 2000 + 1000 = 3000; 400 + 300 = 700; 70 + 50 = 120; 8 + 6 = 14; 3000 + 700 + 120 + 14 = 3834
Add thousands of second addend to first addend, then hundreds; then tens, then ones – ex: 4702 + 3859: 4702 + 3000 = 7702; 7702 + 800 = 8502; 8502 + 50 = 8552; 8552 + 9 = 8561
Compensation (works very easily in some cases) – ex: 3567 + 1997: 3567 + 2000 = 5567; 5567 – 3 = 5564
Standard American algorithm WITH understanding – ex: 2468 + 5967:

∑ 8 + 7 = 15, record the 5, and add the 1 (10) from 15 to the tens column;

∑ 1 + 6 + 6 (which really is 10 + 60 + 60) is 13 (which really is 130), record the 3 (30), and add the 1 (100) to the hundreds column;

∑ 1 + 4 + 9 (which is really 100 + 400 + 900) is 14 (which really is 1400); record the 4 (400), and add the 1 (1000) to the thousands column;

∑ 1 + 2 + 5 (which is really 1000 + 2000 + 5000) is 8 (which really is 8000); answer is 8435

For subtraction:

Subtract thousands, then hundreds, then tens, then ones; then combine – ex: 6394 - 2876: 6000 – 2000 = 4000; 300 – 800 = -500; 90 – 70 = 20; 4 – 6 = -2; 4000 - 500 + 20 – 2 = 3518
Find the distance from one value to the other: ex: 3760 – 2831: 2831 to 2900 is 69; 2900 to 3760 is 860; 860 + 69 = 929
Compensation (works very easily in some cases) – ex: 8357 – 3989: 8357 – 4000 = 4357; 4357 + 11 = 4368 (since taking away 4000 from 8357 was 11 too much)
Standard American algorithm WITH understanding – ex: 8123 - 6579:

∑ Choose not to take 9 from 3, so regroup one ten; 13 – 9 = 4;

∑ Choose not to take 7 from 1 (which is really 70 from 10), so regroup one hundred; 11 – 7 = 4 (which is really 110 – 70 = 40);

∑ Choose not to take 5 from 0 (which is really 500 from 0 hundreds), so regroup one thousand; 10 – 5 = 5 (which is really 1000 – 500 = 500);

∑ Subtract hundreds; 7 – 6 = 1 (which is really 7000 – 6000 = 1000); answer is 1544

For fluency in multiplying and dividing by double-digit numbers and multiples of 10 (grade 4):

The following are extensions of the strategies from grade 3, most of which rely on the distributive property. These are examples and not intended to represent every possible strategy. Also see our videos for demonstrations of many of these strategies. The most important goal is that students should be able to use numbers flexibly and efficiently to compute accurately because one strategy may be very easy to use in one problem and not so easy to use in another.

For multiplication (several of these can be modeled using the areas of rectangles; see third strategy below for an example):

Repeated addition (only to be used initially) – ex: 65 x 12 = 130 x 6 = 130 + 130 + 130 + 130 + 130 + 130 = 780
Decomposing by tens and ones (also used initially) – ex: 27 x 53 = (10 x 53) + (10 x 53) + (7 x 53) = 530 + 530 + 371 = 1431
Decomposing by place value – ex. A: 57 x 82 = (50 x 82) + (7 x 82) = (50 x 80) + (50 x 2) + (7 x 80) + (7 x 2) = 4000 + 100 + 560 + 14 = 4674

∑ Sketch a rectangle with sides 268 and 7 units; divide this into three rectangles: 200 by 7, 60 x 7, 8 x 7; find the area of each; add these areas

Other partitions – ex: 94 x 45 = (100 x 45) - (6 x 45) = 4500 - 270 = 4230
Standard American algorithm WITH understanding – ex: 35 x 78:

∑ 5 x 8 = 40; record the 0; regroup the 4 (really 40);

∑ 3 x 8 = 24 (really 30 x 8 = 240); 24 + 4 = 28 (really 240 + 40 = 280); record the 28 (really 280);

∑ 5 x 7 = 35 (really 5 x 70 = 350); record the 50; regroup the 3 (really 300);

∑ 3 x 7 = 21 (really 30 x 70 = 2100); 21 + 3 = 24 (really 2100 + 300 = 2400); record the 24 (really 2400);

∑ Add: 280 + 2450 = 2730

Multiplying by larger multiples of 10 (such as 300, 1900, etc.): We want students to recognize that multiplying by 300, for instance, is the same as multiplying by 3 and then by 100. In order to do this, they must understand the pattern that when we multiply a whole number by 100, we include 2 extra zeros at the end of it. (ex: 5 x 100 = 500; 9 x 100 = 900) So, to multiply by 300, we would use any of the strategies above to multiply by 3 and then multiply that answer by 100.

For division (the symbol / is used below for the division sign because not all web browsers can read the division sign as a symbol):

Use multiplication/building up – ex: 1263 / 37 (How many groups of 37 are in 1263?)

∑ 37 x 10 = 370; 370 + 370 = 740 (20 groups of 37 so far);

∑ 740 + 370 = 1110 (30 groups of 37 so far);

∑ 37 x 2 = 74; 1110 + 74 = 1184 (32 groups of 37 so far);

∑ 1184 + 74 = 1258 (34 groups of 37 so far)

∑ 1263 – 1258 = 5; so, the answer is 34 groups of 37 with 5 left over

Take out multiples of 10 of the divisor – ex: 2643 / 81 (How many groups of 81 are in 2643?)

∑ 2643 – 810 = 1833 (took out 10 groups of 81);

∑ 1833 – 810 = 1023 (took out 10 more groups of 81);

∑ 1023 – 810 = 213 (took out 10 more groups of 81);

∑ 213 – 162 = 51 (took out 2 more groups of 81);

∑ 10 + 10 + 10 + 2 = 32; so, 32 groups of 81 with 51 left

Modified standard American algorithm – ex: 5961 / 28 (How many groups of 28 are in 5961?) - *Since this is difficult to display as a graphic on a webpage so that all web browsers can interpret it, you may want to jot down the problem in the standard format and follow along with the steps as described below.

∑ Think: How many times does 28 go into 5900? (200) Write 200 above the division sign; subtract 5600 from 5961 to get 361.

∑ Think: How many times does 28 go into 361? (at least 10) Write + 10 after the 200 above the division sign; subtract 280 from 361 to get 81.

∑ Think: How many times does 28 go into 81? (2) Write + 2 after the + 10 above the division sign; subtract 56 from 81 to get 25.

∑ The answer is 212 (200 + 10 + 2) with 25 left over.

Standard American algorithm WITH understanding – ex: 3792 / 41 (How many groups of 41 are in 3792?) - *Since this is difficult to display as a graphic on a webpage so that all web browsers can interpret it, you may want to jot down the problem and follow along with the steps as described below.

∑ Think: How many times does 41 go into 379 (since it cannot go into 37 evenly)? 90 – record the 9 above the 9 (in the tens place to indicate 90), and write 3690 below 3792 (instead of just 369) since 41 x 90 = 3690.

∑ Subtract: 3792 – 3690 = 102.

∑ Think: How many times does 41 go into 102? 2 – record the 2 above the 2 in 3792, and write 82 below 102 since 41 x 2 = 84.

∑ Subtract: 102 – 82 = 20. The answer is 92 with 20 left over.

Dividing by larger multiples of 10 (such as 400, 1200, etc.): We want students to recognize that dividing by 400, for instance, is the same as dividing by 4 and then by 100. In order to do this, they must understand the pattern that when we divide a whole number by 100, we take away two zeros at the end of it (or move the decimal point two places to the left, but this is rarely needed in grade 4). (ex: 600 / 100 = 6; 2400 / 100 = 24) So, to divide by 400, we would use any of the strategies above to divide by 4 and then divide that answer by 100.

Other helpful activities for grade 4, any time, any place:

How much is needed to get to 1000? Give your child a number between 0 and 1000, and ask how much is needed to get to 1000. For instance, if you said 610, your child would say 390. If you said 513, your child would say 487. This is a very helpful element of developing mental computation skills. Most children will have to start with multiples of 100, 10, or 5, but playing the game Close to 1000 will help them start with other numbers also. An appropriate extension for grade 4 would be to start between 0 and 10,000 ask how much is needed to get to the next multiple of 1000.
What is 10 more? 10 less? 20 more? 30 less? 100 more? 400 less? 1000 more? 3000 less? Give your child a number between 0 and 1000, and ask what number would be 10 more than that number (or 10 less, or 20 less, or 50 more, or 100 more, or 200 less, or 1000 more, or 4000 less, or any multiple of 10, 100, or 1000 more or less). This is a great place value activity and also helpful in developing mental computation skills. With four-digit numbers, most children will have to start with 1000 more and 1000 less and later work up to higher values and 10 or 100 more/less (it is a bit harder to find 10 or 100 more/less when the numbers are in the thousands since children tend to focus on the thousands). An appropriate extension for grade 4 would be to ask about (for instance) 2500 more, 4100 less, etc. This activity can also be done in the context of money (what is $1.50 cents more?).

Grade 5

The fluency activities for grades 5 and 6 primarily relate to whole number theory and fractions/decimals/percents. We develop the following strategies and skills with students in grade 5. Many of these are good activities for any time, any place; also see the activities for grade 4 (any time, any place) immediately above.

Whole number theory:

Identifying square numbers and square roots: A square number is the product of a number, the square root, multiplied by itself; square numbers can be represented by a square with the square root being the side length – 25 (5 x 5), 100 (10 x 10), 16 (4 x 4), 49 (7 x 7), etc.
Identifying prime and composite numbers: Prime numbers have exactly two factors, themselves and 1 (7, 19, 2, 43, etc.). That is, only these two values divide into the number evenly. Composite numbers have more than two factors (32, 75, 10, 8, 26, etc.). That is, more than two values divide into any composite number evenly.
Order of operations: When simplifying an expression, we simplify in the following order (moving from left to right within each of these four steps): anything in parentheses, any exponents, any multiplication or division, any addition or subtraction – example: 5 + (6 – 3) x 2 = 5 + 3 x 2 = 5 + 6 = 11

See grade 3 and grade 4 above for strategies we use for developing multiplication and division facts with students. Click here for a Microsoft Word document that lists many fun ways to practice basic facts once students have learned them. See below for some websites that provide fluency practice.

Fractions/decimals/percents:

Benchmark percents: To find 10% of a number, divide it by 10 (since there are ten 10s in 100); to find 25% of a number, divide it by 4 (since there are four 25s in 100); to find 1% of a number, divide it by 100 (since there are one hundred 1s in 100).
Equivalent fractions and percents: Students should understand and commit to memory common equivalents like 1/5 = 20%, 1/3 = 33 1/3%, 3/4 = 75%, and so on; one way to find the percent from the fraction is to divide the numerator by the denominator and multiply by 100.
Rounding decimals: Students may be asked to round to the nearest tenth, hundredth, or thousandth; look at the decimal place to the right and round up if 5 or above; examples: (to tenths) 3.467 becomes 3.5; (to hundredths) 71.243 becomes 71.24; (to thousandths) 8.34178 becomes 8.342.
Comparing decimals: To compare decimals, they must be compared place for place from left to right; for example, 2.3 is larger than 2.165 even though 2.165 appears larger – compare 2 to 2 (equal), 3 to 1 (this is how we know 2.3 is larger); we could go further if needed. Another example: 50.6 is larger than 5.06 because we compare the 5 in 50.6 to nothing in the tens place in 5.06.

Grade 6

The fluency activities for grades 5 and 6 primarily relate to whole number theory and fractions/decimals/percents. We develop the following strategies and skills with students in grade 6, along with reviewing those from grade 5 above. Several of these (from grades 5 and 6) are good activities for any time, any place; also see the activities for grade 4 (any time, any place).

Whole number theory:

Prime factorization: Breaking a number into prime factors: 75 = 3 x 5 x 5 or 3 x 5²; 80 = 2 x 2 x 2 x 2 x 5 or 2⁴ x 5. A prime number is only evenly divisible by itself and 1.
Greatest common factor (GCF): The largest number that is a factor of (evenly divides into) two or more given numbers; the GCF of 30 and 45 is 15 because no larger number fits evenly into both.
Least common multiple (LCM): The smallest number that is a multiple of two or more given numbers; the LCM of 4, 9, and 12 is 36 because they all can be multiplied by another number to get 36.

Fraction, decimal, percent operations:

Please refer to resources from the textbook, class notes, and the sixth grade homework help website for help with these operations.

Some websites that can provide fluency practice (Stow-Munroe Falls City Schools assumes no responsibility for the content of these sites or links from them):

http://www.mathfactcafe.com

http://www.math.com

http://www.harcourtschool.com/menus/math2002/na/menu_na.html

http://www.funbrain.com

http://www.bbc.co.uk/education/mathsfile/gameswheel.html

http://www.bbc.co.uk/schools/ks2bitesize/maths/

http://www.aplusmath.com/

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