Description: Continue to link abstract level instruction to contexts used to teach the skills at the concrete and representational/drawing levels.
Purpose: to assist students to build meaningful connections between what they know about rounding to the nearest ten and/or hundred by drawing to rounding numbers without drawing pictures.
Learning Objective 1: Round to the nearest ten or hundred by identifying the talking digit in two and three digit numbers.
Materials:
Teacher –
One example of the rounding process using concrete materials and a number line that can be clearly seen by all students.
One example of the rounding process using drawings and a number line that can be clearly seen by all students.
A written learning objective that is clearly visible to all students: “Use the TALKING DIGIT to round numbers to the nearest ten or hundred.”
Description:
1.) L ink to students’ prior knowledge of rounding to the nearest ten and hundred using concrete materials, by drawing, and using a number line with drawings.
For Example:
You have learned to round to the nearest ten and to the nearest hundred by using three very helpful strategies. You first learned how to round by using concrete materials like these baseten materials (Hold up/display an example of rounding with concrete materials.). You also have learned how to draw pictures to help you round to the nearest ten and hundred (Display an example of rounding a number with drawings and a number line.) Last, you learned how to use both concrete materials and drawings to round using a number line. (Point to the number line in both the concrete and drawing examples.)
2.) I dentify the skill students will learn: “Round to the nearest ten or hundred by identifying the talking digit in two and three digit numbers.”
For Example:
Today, we are going to learn how to round to the nearest ten and hundred by using a number line without concrete materials or drawings. We are going to learn something special about the digits in the number we want to round. I will show you how to use the “Talking Digit” to help us round numbers to the nearest ten or hundred. (Display the written learning objective and point to it as you say it.) What are we going to learn today? (Point to the written learning objective and elicit the response, “Use the TALKING DIGIT to round numbers to the nearest ten and hundred.”) That’s right, we’re going to learn how to use the talking digit to round numbers to the nearest ten and hundred.
3.) P rovide rationale/meaning for rounding to the nearest ten and hundred.
For Example:
Learning how to use the talking digit in a number to round will really help you to round when you don’t have concrete materials and when you don’t have time to draw in order to round a number. You will not always have concrete materials when you need to round and there will be times when drawing will take too long. What are some examples of times we might need to round a number to its nearest ten or hundred? (Prompt students to offer previous contexts used to teach at the concrete and representational levels and to offer “new” examples.)
Purpose: to provide students a clear teacher model of rounding to the nearest ten and hundred without using concrete materials and without drawing.
Learning Objective 1: Round to the nearest ten or hundred by identifying the talking digit in two and three digit numbers.
Materials:
Teacher –
An appropriate format to display numbers and number lines so that all students can see and hear.
A selection of three digit numbers that are color coded (e.g. ones place is green, tens place is red, and hundreds place is blue – 347.)
A language card with the following phrase written: “round to the nearest HUNDRED.”
Number lines prepared that represent the appropriate “hundreds” the given three digit number might be rounded to. The numbers represented should be increments of ten and the tens digit should be colorcoded appropriately.
For Example:
Description:
A. Break down the skill of rounding to the nearest ten or hundred by identifying the talking digit in two and three digit numbers.
1.) Read the number.
2.) Identify whether you are rounding to the “ten” or to the “hundred.”
3.) Identify the “talking digit” and underline it.
4.) Use number line and 'talking digit' to round to nearest ten or hundred.
5.) Review what the "talking digit" is and how it helps when you need to round..
Learning Objective 2: Round by writing numbers to represent “counting on” and “counting back” to the nearest ten or hundred.
Materials:
Teacher –
Chalkboard, dryerase board, chart paper, or overhead to write on.
Multicolored chalk, markers or pens for writing.
Description:
A. Break down the skill of rounding by writing numbers to represent “counting on” and “counting back” to the nearest ten or hundred.
1.) Identify whether you are rounding to the nearest ten or hundred and decide what the “greater” and “lesser” ten or hundred is.
2.) Identify the talking digit of the number to be rounded and decide whether the “talking digit” tells you to “count on” and “count back” by ones or by tens.
3.) Count on by writing numbers (by ones for rounding to the nearest ten and by tens for rounding to the nearest hundred.).
4.) Count back by writing numbers (by ones for rounding to the nearest ten and by tens for rounding to the nearest hundred.).
5.) Compare the “count on” set of numbers to the “count back” set of numbers and decide which ten or hundred the number should be rounded to.
6.) Write the nearest ten or hundred.
Learning Objective 3: Rounding to the nearest ten or hundred using proximity cues (Round to the Nearest Ten Chart or Round to the Nearest Hundred Chart).
*The Round to Nearest Ten Chart and Round to Nearest Hundred Chart provide students a clear teacher model of how to use proximity as a cue for rounding.
Materials:
Teacher –
A “Round to Nearest Ten Chart” (i.e. a chart that depicts the numbers 0 –100 grouped by tens in rows so that each row depicts the “lesser” and “greater” ten:
For Example:
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A “Round to Nearest Thousand Chart” (i.e. a chart that depicts multiples of ten from 01000 grouped by ten multiples in rows so that each row depicts a the “lesser” and “greater” hundred.
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Nine “peep hole” cards. Each card is one row wide and nine columns long. Each card has one square cutout space positioned for one number position in columns 29 (e.g. one card when positioned between the first and last number in a row will reveal the second number in the row; the second card when positioned between the first and last number in a row will reveal the third number in the row; etc.
For Example:
*This “peep hole” card reveals the “third” number in a row when placed between the first and last number in the row.
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A ruler or tape measure
Description:
A. Break down the skill of rounding to the nearest ten or hundred using proximity cues (Round to Nearest Ten Chart and Round to Nearest Hundred Chart).
1.) Introduce the Round to Nearest Ten Chart and/or Round to Nearest Hundred Chart.
2.) Identify a number to be rounded and whether it will be rounded to the nearest ten or hundred.
3.) Identify which chart can be used for rounding to the nearest ten (Round to Nearest Ten Chart) and which chart can be used for rounding to the nearest hundred (Round to Nearest Hundred Chart).
4.) Identify the “talking digit” and what number on the Round to Nearest Ten Chart or Round to Nearest Hundred Chart the “peep hole” card should be placed.
5.) Choose the appropriate “peep hole” card to use and place it between the “lesser” and “greater” tens or the “lesser” or “greater” hundreds.
6.) Model determining which ten or hundred to round to by pointing out the distance between the number you are rounding and each ten or hundred.
7.) Repeat this process at least three times for both rounding to the nearest ten and to the nearest hundred.
Purpose: to provide students the opportunity to build their initial understanding of how to round numbers to the nearest ten/hundred without concrete materials or drawings, and to provide you the opportunity to evaluate your students’ level of understanding after you have initially modeled this skill.
Materials:
*Dependent on the skill you are Scaffolding Instruction for (See the materials listed for the specific skill you want to scaffold under Explicit Teacher Modeling).
Description:
*Scaffolding at the abstract level of instruction should occur using the same process as scaffolding instruction at the concrete and representational/drawing levels of instruction (See the description of Scaffolding Instruction for, “rounding to the nearest hundred using base ten materials and a number line,” in the Concrete Level Instructional Plan.). The steps listed for each skill during Explicit Teacher Modeling should be used as structure for scaffolding your instruction.
A. Scaffold instruction using a high level of teacher direction/support. (*Dependent on the needs of your students, you may want to continue to associate drawings to the abstract level rounding process during this phase of scaffolding. Move to the next phase of scaffolding only when students demonstrate understanding and ability to respond accurately to your prompts.
B. Scaffold instruction using a medium level of teacher direction/support. (If you associated drawings with the abstract process for rounding while scaffolding using a high level of teacher direction/support, then do not include drawings during this phase of scaffolding). Move to the next phase of scaffolding only when students demonstrate understanding and ability to respond accurately to your prompts.
C. Scaffold instruction using a low level of teacher direction/support. (Students should actually round as you prompt them during this phase of Scaffolding Instruction.) Move students to independent practice of the skill only after they demonstrate the ability to perform the skill with limited prompting from you.
Purpose: to provide students multiple opportunities to round to the nearest ten or hundred by writing numbers (instead of drawing pictures) to “count on” and “count back” to round to the nearest ten or hundred..
Learning Objective 3: Round to the nearest ten/hundred by drawing using a number line.
Instructional Game – Board Game
Materials:
Teacher –
Generic game boards (*Game boards can be made by using manila folders or square pieces of tagboard and configuring multiple spaces (i.e. 1530 spaces will provide multiple practice opportunities) on which students can move their game pieces. A rectangle can be drawn on the gameboard to represent where the response cards are placed. (At the beginning of the year, students can be taught how to make game boards and then these gameboards can be used throughout the year!) Gameboards can be as colorful/creative or as simple in nature as appropriate given your time and your student’s ability level. Old or discarded commercially made game boards also can be used (e.g. “Monopoly,” “Life,” “Stratego,” etc.)
Dice or spinners
Multiple sets of cards that have: Front  1.) the number to be rounded and the numbers representing counting on/counting back to the nearest ten/hundred written on either side (e.g. 30 21 32 33 34 35 36 37 38 39 40); 2.) below the example is written one or more questions with three choices; Back – 1.) the correct choice(s) is written. *To facilitate making these cards, you can make one set for each level of difficulty using plain paper so that the examples, questions, and choices are a size that can fit on one side of a 4x5 notecard. Number each example #1  #20 in the top right corner so that each card has the number appear at the top right corner. Then you can make multiple copies of the one set, cut them out, paste them on notecards, and laminate the notecards. Students can assist in pasting the drawings/choices & answers on notecards!
A copy of the original set of problems so the teacher can check individual student response sheets.
Students 
Each small group has a gameboard, die or spinner, a set of cards.
A sheet of paper to record which number example they respond to and whether they answered it correctly or not.
Description:
Activity:
Students play in small groups using a generic game board (See description under “Materials.”) Students respond to cards that depict the number to be rounded in a dark color (black or blue) and then numbers that count on and count back to the nearest ten or hundred in a contrasting color (red). A simple question is written below the example with three choices (e.g. What is the nearest ten? How many ones to the greater ten?” How many tens to the lesser hundred?). On the back is the correct answer. To move, students roll a die or spin a spinner. If they respond correctly, then they move the appropriate number of spaces. To evaluate student performance, students can record the question number they respond to on a sheet of paper and make a mark indicating whether they answered it correctly or incorrectly. To add more challenge, cards can be divided into two or three piles that represent more and more challenging questions. Students choose which level they want to answer. Students can be rewarded for answering more challenging questions correctly by moving one or more additional spaces.
Instructional Game Steps:
1.) Introduce game.
2.) Distribute materials.
3.) Provide directions for game, what you will do, what students will do, and reinforce any behavioral expectations for the game.
4.) Provide time for students to ask questions.
5.) Model how to respond to the card prompts.
6.) Provide time for students to ask questions about how to respond.
7.) Model how students can keep track of their responses.
8.) Play one practice round so students can apply what you have modeled. Provide specific feedback/answer any additional questions as needed.
9.) Monitor students as they practice by circulating the room, providing ample amounts of positive reinforcement as students play, providing specific corrective feedback/ remodeling skill as needed.
11.) Play game.
12.) Encourage students to review their individual response sheets, write the total number of “correct” responses under the “C” (Correct) column and do the same for the “H” (Help) column.
13.) Review individual student response sheets to determine level of ../../../understanding/proficiency and to determine whether additional modeling from you is needed.
Purpose: to provide students multiple opportunities to “solidify” connections between what they know about rounding to the nearest ten and hundred by drawing to rounding to the nearest ten and hundred at the abstract level. The use of a structured “planned discovery activity sheet” provides students who have learning problems appropriate cueing that allows them to independently make the connections between their “representational/drawing level” of understanding and their abstract level of understanding.
Learning Objective 1: Round to the nearest ten or hundred by identifying the “talking digit” in two and three digit numbers.
1. Planned Discovery Activity
Materials:
Teacher –
Appropriate number of structured “planned discovery" activity sheets. The sheet contains multiple examples of the following: 1.) a drawing of rounding to the nearest ten or hundred (e.g. a number line with the appropriate drawings above that represent the number to be rounded, the “count on” drawings, and the “count back” drawings.); 2.) the following prompts/questions with a space provided to write the player’s answer: What number is being rounded? Rounding to nearest ten or hundred? What is the talking digit? What ten or hundred should number be rounded?
Example for rounding “162 to nearest hundred:”
What number is being rounded? _____________ Rounding to the nearest ten or hundred? ______________
What is the talking digit? _________________ What ten or hundred should number be rounded? _________
Answer key for planned discovery activity sheet.
Cue cards that show one example of rounding to tens and one example of rounding to hundreds with appropriate answers to the questions/prompts. Students who need additional prompting can use these cards as needed to respond to the planned discovery activity sheet.
Students 
Each pair has two planned discovery activity sheets.
“Example” cue cards if appropriate.
Pencils for writing.
Description:
Activity:
Students work in pairs to respond to a planneddiscovery learning sheet (See description under “Materials.”). Each sheet has drawings on a number line that represent rounding to the nearest ten or hundred (the numerical form of the value to be rounded is not written). Students respond by writing the number that is being rounded, identifying whether the drawing represents rounding to the nearest ten or hundred, identifying the “talking digit,” and identifying what ten or hundred the number should be rounded to. Students take turns responding to each example. The “coach” describes the drawings and then asks the “player” to respond to each question/prompt for that drawing. The coach writes the player’s answer in the appropriate space on that student’s planneddiscovery learning sheet. The coach and player then refer to the answer key to check the player’s responses. If the player’s responses are correct, the students switch “roles” and move to the next example. If the player’s responses are incorrect, the coach and player discuss why the player’s response differed from the correct answer. After they have reached agreement, the students raise their hand to signal the teacher. When the teacher approaches them, the player (or coach and player) explain what they learned (why the response was not correct). For students who may need additional prompting/cueing, two example cards could be provided. One “example” could show a drawing representing rounding to tens with appropriate responses to the questions and one “example” could show a drawing representing rounding to hundreds with appropriate responses to the questions. Students could refer to these “example” cards as needed as they respond. The teacher circulates the room monitoring students as they work, providing positive reinforcement, specific corrective feedback, and listening to student explanations.
Planned Discovery Activity Steps:
1.) Develop Planned Discovery Activity Learning Sheet as described under Materials.
2.) Distribute the Planned Discovery Activity Learning Sheet and provide clear directions for completing the activity, including appropriate behavioral rules.
3.) Model how to respond to one example on the Planned Discovery Learning Sheet (and model appropriate behaviors as needed).
4.) Provide students with appropriate materials (e.g. Cue cards).
5.) Monitor students as they practice, providing appropriate corrective feedback, prompting student thinking, providing positive reinforcement, and modeling or cueing as needed.
6.) At the conclusion of the activity, provide students with solutions to the Planned Discovery Activity Learning Sheet. Emphasize why the answers are correct.
7.) Review student response sheets and note special difficulties individual students may be having and/or progress they are making.
Purpose: to provide you with continuous data for evaluating student learning and whether your instruction is effective. It also provides students a visual way to “see” their learning.
Materials:
Teacher –
Appropriate prompts if they will be oral prompts
Appropriate visual cues when prompting orally
Student –
Appropriate response sheet/curriculum slice/probe
Graph/chart
Description:
Steps for Conducting Continuous Monitoring and Charting of Student Performance:
1.) Choose whether students should be evaluated at the receptive/recognition level or the expressive level.
2.) Choose an appropriate criteria to indicate mastery.
3.) Provide appropriate number of prompts in an appropriate format (receptive/recognition or expressive) so students can respond.
At the abstract level of understanding, the most efficient format for a curriculum slice/probe is written (e.g. student responds in writing to written prompts). In some cases, you may want to use oral prompts where written examples are provided on the chalkboard/dryerase board or overhead projector (e.g. three digit numbers written above various number lines and students respond to several teacher questions about each example: “What is the nearest ten? What is the nearest hundred? How many ones did you count on to the greatest ten? How many tens did you count back to the lesser hundred?).
4.) Distribute to students the curriculum slice/probe/response sheet/concrete materials.
5.) Give directions.
6.) Conduct evaluation.
7.) Count corrects and incorrects/mistakes (you and/or students can do this depending on the type of curriculum slice/probe used – see step #3).
8.) You and/or students plot their scores on a suitable graph/chart. A goal line should be visible on each students’ graph/chart that represents the proficiency (near 100% accuracy with two or fewer incorrects/mistakes) and a rate (# of corrects per minute) that will allow them to be successful when using that skill to solve reallife problems and when using the skill for higher level mathematics that require use of that skill.
9.) Discuss with children their progress as it relates to the goal line and their previous performance. Prompt them to selfevaluate.
10.) Evaluate whether student(s) is ready to move to the next level of understanding or has mastered the skill at the abstract level using the following guide:
Abstract Level: demonstrates near 100% accuracy (two or fewer incorrects/mistakes) and a rate (# of corrects per minute) that will allow them to be successful when using that skill to solve reallife problems and when using the skill for higher level mathematics that require use of that skill.
11.) Determine whether you need to alter or modify your instruction based on student performance.
*This assessment activity can be used with students who demonstrate difficulty with rounding at the abstract level.
A. Flexible Math Interview/CRA Assessment
Purpose: to assess where student understanding of the rounding process is “breaking down.”
Materials:
Teacher –
Appropriate concrete materials for rounding (See Concrete Level Instructional Plan – Explicit Teacher Modeling.).
Number lines for rounding to nearest ten or hundred.
Appropriate examples for assessment (nearest ten and nearest hundred)
Paper to record notes.
Description:
Have students round to the nearest ten and hundred using concrete materials, by drawing, and without concrete materials or drawings. Also have students round with and without the use of number lines. Ask students to explain their answers as they respond. Note where in the rounding process students “break down;” both at what level they begin having difficulty and at what point within that level of understanding they demonstrate mis../../../understanding/nonunderstanding. Based on where students demonstrate difficulty, provide explicit teacher modeling at that level of understanding and for the particular subskill they are having difficulty with. As the student demonstrates understanding, scaffold your instruction until they are ready to practice the skill independently. As students demonstrate mastery of the skill at that level of understanding, then provide explicit teacher modeling at the next level of understanding. Follow this process until students demonstrate mastery at the abstract level.
Key Ideas
Students who demonstrate difficulty at the abstract level of understanding may have “gaps” in their understanding that can trace back to the representational/drawing level or even the concrete level. By providing additional teacher modeling at the level their “gap” in understanding began and then moving them from a concretetorepresentationaltoabstract level of understanding, you can assist students to become successful at the abstract level of understanding.
Sometimes students demonstrate difficulty at the abstract level because they did not receive enough practice opportunities at the concrete and representational/drawing levels. The drawing level is a very important step for these students. Some students need continued practice drawing solutions and associating their drawings to the abstract symbols and the mental processes necessary to perform at the abstract level.
Some students understand the concept, but have difficulty remembering the steps involved to perform the skill at the abstract level. Providing students with cues they can refer to as they practice at the representational/drawing and abstract levels of instruction is very helpful. Such cueing provides them the independence to practice. Multiple practice opportunities translate into repetition and repetition enhances memory.
*See suggestions described under this section for both the concrete and representational level instructional plans.
