decimals

Rounding activities (whole numbers and decimals)

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by C. Elkins, OK Math and Reading Lady

Last week I reposted my blog regarding use of number lines to assist students with number sense and rounding. Check it out for free activities and rounding charts. Today I am sharing some more rounding activities I developed and used with students to practice (with either whole numbers or decimals). These activities can be varied to suit your students’ needs.

These grid templates are to use the activities with 2-4 students (or teacher vs. student if working one-on-one online). I developed 3 different grid sizes (4 x 4, 5 x 5, and 6 x 6).  You will also need something to generate numbers for each set of players:

  • Grid for playing board:  Get here FREE  Grid 4 x 4   Grid 5 x 5   Grid 6 x 6
  • 2 dice (1-6)
  •  2 dice (1-9)
  • digit cards (0-9) — get your free set here:  0-9 digit cards
  • deck of playing cards (with tens and face cards removed)
  • spinner (with digits 0-9) — 1 is ok, 2 is better

The objective of the game is for a player to capture 4, 5, or 6 squares in a row (horizontally, vertically, diagonally).  You decide based on the size of the grid and the skill level of the players how many captured squares are needed.

The teacher can write in possible answers on the grid and laminate for continued use (samples below). Then students can use a game piece  (flat stones, two-color counters, etc.) or different color dry erase marker to mark their square.

  • Using a paper form, students can write in answer choices randomly on the grid (supplied by the teacher for accuracy). Then each player can use a different colored crayon to mark their square.

Here are some different variations of the game (whole number rounding to nearest 10, 100, 1000 and decimal rounding to the nearest tenth or hundredth).

Rounding to the nearest ten:  You can use the blank grid to write in your own numbers randomly.  Consider which number generated options you are using.  If you use 1-6 dice, the biggest number on the board has to be 70 and remember there’s only 2 ways to achieve 70 (by rolling a 6 and 5 or a 6 and 6).  If you use 1-9 dice or number cards, then you can place numbers from 10-100 on the board.  This gives a few more options and a chance to round higher numbers.

  • Roll 2 dice (or turn over 2 number cards, spin spinner twice)
  • Generate a 2 digit number.  If a 3 and 5 are rolled, the player can decide to make it 35 or 53.
  • Round that number to nearest 10.
  • Find that number on the grid.
  • If using a laminated board, place a colored “chip” on it. If using paper, each player colors their chosen # with a crayon.
  • Player #2 follows same steps.
  • Each player is trying to get 4, 5, or 6 in a row (depending on which grid size you choose).
  • It’s more fun if you try to block the other player and use strategies about your choice of a number to round (should I use 35 — rounded to 40?  Or 53 — rounded to 50?)

Rounding to the nearest hundred:

  • Follow same steps as above, except use 3 dice or 3 number cards.
  • Place numbers such as 0, 100, 200, 300 . . . randomly on the board. In the samples pictured I numbered to 1000 since I used 0-9 dice. I didn’t show a 0 on the boards pictured below, but should have since a number less than 50 could actually be generated. If using 1-6 numbered dice, the highest would be 700.
  • Example:  Roll a 2, 5, 6 — player can make these numbers 256, 265, 526, 562, 625, 652.  The number choice becomes part of the strategy of the game to see which spot is available on the board.

Rounding to the nearest thousand:

  • Follow same steps as above, except use 4 dice or 4 number cards.  If using 1-6 numbered dice, the highest would be 7000.

Rounding to the nearest tenth:

  • Follow steps similar to rounding to nearest tenth, except answer choices on the grid would look like this:  .1, .2, .3 . . .
  • If using number cards (as pictured below) or a spinner with digits up to 9, be sure to include a space on the grid for 1 (which is what you would round these numbers to:  .95, .96, .97, .98, or .99.
  • Again, be mindful of randomly placing numbers because it depends on which number generating options you are using.  If using 1-6 dice, I would only include a couple of spaces with .7 because there’s a limited number of ways to round to .7 with dice numbered 1-6.  The only way to round to .7 would be to roll a .65 or a .66.

Rounding to the nearest hundredth:

  • Follow steps similar to rounding to the nearest hundred by using 3 dice or turning over 3 number cards.  Be sure to include a space or two for an answer of 1.

Other tips for playing:

  1. Provide students with a blank white board to draw an open number line to check out their answer.
  2. Provide a sentence frame such as:  I made the number  ______ which is rounded to ________.
  3. Remind the players that it is their job to watch their opponent and challenge anything they think may not be correct (in a friendly, helping manner of course).
  4. Shorter time frame for playing?  Choose the 4 x 4 grid.  Longer time frame?  Choose the 6 x 6 grid or use the 6 x 6 grid with the winner being one to get 5 in a row.
  5. Consider creating a box of 4 completed squares in addition to 4 in a row.
  6. This can be played as teacher vs. students in a virtual setting.
  7. This can be played in a one-on-one online setting by using a document camera or posting a screen shot on the screen.

Let me know if you try these!  Pass along any extra tips you have.

Also, a reminder to contact me if you would like personalized professional development over any reading or math strategy.  I can do a Zoom session with you or a group of teachers.  Flexible payment options.  Also, check out my link on the side bar for Varsity Tutors regarding the opportunity for you to tutor students online or in person (and earn a bonus for using my name).

Take care, stay safe!!!  

 

Rounding and Number Lines

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by C. Elkins, OK Math and Reading Lady 

I get requests from many teachers to help with instructional strategies regarding rounding, so I am happy to share my thoughts (and freebies) with you. Difficulty with rounding usually means students lack number sense. The essential goal of rounding is: Can you name a benchmark number (whole, tens, hundreds, thousands, tenths, hundredths, etc.) that a given number is closer to? I have found the more experience a student has with number lines, the better they will be with number sense, and the better they are with rounding to the nearest ___.  Then this rounding practice must be applied to real world problems to estimate sums, differences, products, or quotients.

When doing a google search for tips on rounding (ie Pinterest), you very often find an assortment of rhymes (such as “5 or more let it soar, 4 or less let it rest”) and graphics showing underlining of digits and arrows pointing to other digits. These steps are supposed to help children think about how to change (or round) a number to one with a zero. Many students can recite the rhyme, but then misunderstand the intent, often applying the steps to the wrong digit, showing they really don’t have number sense but are just trying to follow steps.

My answer (and that of other math specialists) is teaching students how to place any number on a number line, and then determining which benchmark number it is closest to. Continue reading to see examples and get some free activities. And watch next week for some new rounding activities for grades 2-6 (whole numbers and decimals).

Continue reading →

Discovering Decimals Part 2: Addition & Subtraction

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by C. Elkins, OK Math and Reading Lady

Last week, we looked at some ways to gain number sense about decimals. This post will address using decimals in the operations of addition and subtraction . . . and how to model concretely and pictorially. You can also download the color grid pages along with a free decimal math game in this post. Part 3 (future post) will address multiplication and division of decimals.

If you missed last week’s post, please review it first before continuing with this one. Before performing various operations with decimals, students must have a basic understanding of how to represent them concretely, pictorially and numerically.  Example:  .8 = .80 can be proven with base ten blocks and with 100 grid drawings. This understanding should also be linked to fractions: 8/10 is equivalent to 80/100. Click here for pdf of Representing Decimals page.

Addition  

For concrete practice, use a 100 base ten block to represent the whole (ones), the tens rod to represent tenths, and unit blocks to represent hundredths. Construct each addend and then combine them. Ten tenths’ rods become one whole. Ten hundredths cubes become one tenth.

In a pictorial model, shade in the ones, tenths and hundreds on hundred grids. Use different colors to represent each addend. Click here for pdf of Adding Decimals page.

The concrete and pictorial models will also prove .8 = .80, reinforcing the concept that adding a zero to the right of a decimal does not change its value. This will be an important factor when moving to the standard algorithm vertical addition model.

Using an open number line is also a good pictorial model to use when adding decimals, especially if students are already familiar with its use regarding addition of whole numbers. This method reinforces number sense of the base ten system because you continually think, “What goes with .07 to make a tenth?” (answer = .03); or “What goes with .9 to make a whole?” (answer: .1).

A method called partial sums should also help students gain place value number sense along with addition of decimals. Each step is broken down (decomposed). 

Estimating will also be a critical addition problem solving step focusing on number sense. If adding 34.78 plus 24.12, does the student realize their answer should be somewhere close to 35 + 24? Are they thinking, “Is my first number closer to 34 or 35?”  What is halfway between 34 and 35 (34.5 or 34.50)? Where would 34.78 fall if it was on a numberline between 34 and 35? (See my post on rounding for more details.) Continue reading →

Decimals: Part 1 – The Basics (revised)

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by C. Elkins, OK Math and Reading Lady

Number sense regarding decimals usually starts with fourth grade and continues with more complex operations involving decimals in fifth grade and beyond. It is this extension of the place value system and then relating them to fractions and percentages that often perplex our students (and the teachers, too)!  Read ahead to get your freebies (Decimal practice notes, anchor charts, and Discovering Decimals Number of the Day / Game activity).  I have revised this previous post and included some more freebies below.

Students must understand  this base-ten value system extends in both directions — between any two values the 10-to-1 ratio remains the same. When using place value blocks in primary grades, students recognize the 100 square as 100, the tens strip as 10, and the units cube as 1.  Then with decimals, we ask them to reverse their thinking as the 100 square represents 1 whole, the tens strip represents a tenth, and the unit cube represents a hundredth.  This may take repeated practice to make the shift in thinking — but don’t leave it out. Remember the progression from concrete (hands-on) to pictorial to abstract is heavily grounded in research. Students will likely gain better understanding of decimals by beginning with concrete and pictorial representations.

I am sharing my decimal practice notes, which highlight some of the basic concepts to consider when teaching. Pronouncing the names for the decimals is not in these notes, but be sure to emphasize correct pronunciation — .34 is not “point three four.” It is “thirty-four hundredths.” Use the word and for the decimal point when combining with a whole number.  Example: 25.34 is pronounced “Twenty-five and thirty-four hundredths.” I know as adults we often use the term “point,” but we need to model correct academic language when teaching. You can get also the pdf version of these notes by clicking here: Decimal practice teaching notes. Continue reading →

Graphic Organizers for Math

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by C. Elkins, OK Math and Reading Lady

Here are some cool graphic organizers for your math files!  Make sets of them, laminate or put in plastic sleeves, and use them over and over again!  Graphic organizers help students stay organized and teach them how to complete problems neatly. They are also a great way for students to show different strategies for the same problem. While primary students may need an already-made graphic organizer, intermediate students should be taught how to duplicate them on their own to use whenever the need arises – so the simpler, the better! With repeated use, students are more likely to utilize them regularly in their daily work (and on their scratch paper with assessments).

This one has ten frames and part-part-whole models. In my opinion, these are essential when working with K-2 students because they help children with subitizing, number bonds, and addition / subtraction facts.  If you are using Saxon, you are missing these important strategies!!:

Here’s one to show fractions (area, set, length models)

Need a template for students to make arrays? This one is ready!  I love showing students how to break an array into smaller parts to see how multiplication (or division) facts can be decomposed.  Example:  Make a 6 x 7 array.  Section off a 6 x 5 part. Then you have a 6 x 2 part left over.  This proves:  6 x 7 = (6 x 5) + (6 x 2).  Or — 6 x 7 = 30 + 12 = 42

This graphic organizer shows 5 different multiplication strategies using 2 digit numbers, and a blank one for students to record their thinking. Very handy!!  One of my favorite strategies is partial products. I highly recommend this one before going to the std. algorithm because students decompose the problem by place value and must think about the whole number and not just the parts.

Do your students need something to help them see the different models for a decimal? Try out this graphic organizer. Students will utilize the pictorial forms as well as the abstract.

Do your students know that .7 (or 7/10) is the same as .70 (or 70/100)?  Using this dual set of tenths and hundredths grids will help them see why this is true!

Be sure to check out my FREE templates and organizers (see black bar above “links . . .”)  Please share your favorite graphic organizers for math!  Enjoy!!

Discovering Decimals Part 3: Multiplication and Division

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by C. Elkins, OK Math and Reading Lady

This is the last part of number operations dealing with decimals: multiplication and division in a concrete and pictorial method. This is actually not part of the KG-5th grade OAS standards, but it is addressed in 6th grade and for those of you utilizing the CCSS, you will find multiplication and division with decimals starts in 5th grade.  Or you may have advanced students who are ready to explore this concept. There are a couple of freebies included in this post. Read on to find them!

Multiplying Decimals:

Typically we teach our students that when you multiply 2 numbers together, the product is larger than the 2 factors. And when we divide two numbers, the quotient is smaller than the dividend.

Be careful about stating this generalization: This remains true when multiplying whole numbers (or even a combination of whole and decimal such as 5.2 x 6.4 = 33.28 in which the product is larger than either factor), but NOT with decimals or fractions less than 1 (example: .7 x .2 = .14 in which .14 is less than either factor .7 or .2). With division of decimals less than 1, the answer is often a whole number larger than either the divisor or the dividend (such as 5 divided by 1/2 = 10). This is a difficult concept, but modeling and practicing with concrete and pictorial models helps to see the reasoning. So, be careful not to say, “When you multiply two numbers together you always get a bigger number.” because it’s not always true.

To further understand this principal, let’s look at the use of the times (x) sign with whole numbers. The problem 5 x 2 could be shown in an array as 5 rows with 2 in each row (phrased as 5 “rows of” 2).  It can be shown in a set model such as 5 groups and each group has 2 (phrased as 5 “groups of” 2) . It can also be shown in an area model (box) as a shape divided equally into 5 rows and 2 columns (phrased as 5 “by” 2). Continue reading →

Discovering Decimals Part 2: Addition & Subtraction

Posted by

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by C. Elkins, OK Math and Reading Lady

Last week, we looked at some ways to gain number sense about decimals. This post will address using decimals in the operations of addition and subtraction . . . and how to model concretely and pictorially. You can also download the color grid pages along with a free decimal math game in this post. Part 3 (future post) will address multiplication and division of decimals.

If you missed last week’s post, please review it first before continuing with this one. Before performing various operations with decimals, students must have a basic understanding of how to represent them concretely, pictorially and numerically.  Example:  .8 = .80 can be proven with base ten blocks and with 100 grid drawings. This understanding should also be linked to fractions: 8/10 is equivalent to 80/100. Click here for pdf of Representing Decimals page.

Addition  

For concrete practice, use a 100 base ten block to represent the whole (ones), the tens rod to represent tenths, and unit blocks to represent hundredths. Construct each addend and then combine them. Ten tenths’ rods become one whole. Ten hundredths cubes become one tenth. Continue reading →

Discovering Decimals Part 1: Basic Concepts

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by C. Elkins, OK Math and Reading Lady

Number sense regarding decimals usually starts with fourth grade and continues with more complex operations involving decimals in fifth grade and beyond. It is this extension of the place value system and then relating them to fractions and percentages that often perplex our students (and the teachers, too)!  Read ahead to get your freebies (Decimal practice notes, anchor charts, and Discovering Decimals Number of the Day / Game activity).  I have revised this previous post and included some more freebies below.

Students must understand  this base-ten value system extends in both directions — between any two values the 10-to-1 ratio remains the same. When using place value blocks in primary grades, students recognize the 100 square as 100, the tens strip as 10, and the units cube as 1.  Then with decimals, we ask them to reverse their thinking as the 100 square represents 1 whole, the tens strip represents a tenth, and the unit cube represents a hundredth.  This may take repeated practice to make the shift in thinking — but don’t leave it out. Remember the progression from concrete (hands-on) to pictorial to abstract is heavily grounded in research. Students will likely gain better understanding of decimals by beginning with concrete and pictorial representations.

I am sharing my decimal practice notes, which highlight some of the basic concepts to consider when teaching. Pronouncing the names for the decimals is not in these notes, but be sure to emphasize correct pronunciation — .34 is not “point three four.” It is “thirty-four hundredths.” Use the word and for the decimal point when combining with a whole number.  Example: 25.34 is pronounced “Twenty-five and thirty-four hundredths.” I know as adults we often use the term “point,” but we need to model correct academic language when teaching. You can get also the pdf version of these notes by clicking here: Decimal practice teaching notes.

Anchor charts are excellent ways to highlight strategies in pictorial form. Here are some examples of anchor charts to help students relate decimals to fractions, location on a number line, word form, and equivalencies. Get the free pdf version here: Discovering decimals anchor charts. It includes a blank form to create your own.

 

In this model, I chose the 1000 cube to model 356 thousandths. It’s a little tricky – be sure to see that the 300 part is shaded all the way (front and top – picture 3 slices of 100), the 50 part is shaded (front and half the top – picture half of a 100 slice), and the 6 part is just shaded in the front (picture 6 individual parts). The entire cube would represent 1 whole.

Here’s a matching activity / game in which students match decimal to fraction, word form, expanded form, money, and pictorial form. Included is a blank page so you can make your own or have students take notes. Click here for the FREE activity:  Decimal, Fraction, & Money Match

Another resource ($2.50 at TPT from Joanne Miller) to help students relate the decimal to the pictorial form:Decimal 100 grid Scoot

Finally, below is an activity to practice or reinforce decimal concepts. The page showing can be used as a “Number of the Day” practice. I also created a game using this model, and the whole packet is included in this free pdf. Click here: Discovering Decimals number of the day and game

For more teaching help (videos and interactive models) for decimals, check out the following 3 free resources. These are also listed in my resources section of the blog (top black bar):

  1. www.learnzillion.com
  2. www.teachingchannel.org
  3. http://illuminations.nctm.org

As always, you are welcome to share your decimal discovery ideas. Just click the comment box speech bubble at the top of the article or the comment box at the end of the article.