2012-13 and 2013-14 Transitional Comprehensive Curriculum
Grade 3
Mathematics
Unit 2: Place Value and Number Representations
Time Frame: Approximately four weeks
Unit Description
The focus of this unit is the extension of place value concepts and the development of various representations for numbers including inequalities. Applying place value concepts to addition and subtraction is emphasized. Computational strategies including estimation, mental math, use of calculators, and use of paper/pencil are explored.
Student Understandings
Students develop an understanding of place value, comparing and ordering numbers, rounding, and adding and subtracting 3-digit numbers. Students apply appropriate strategies to a given situation and use them to solve problems.
Guiding Questions
1. Can students determine appropriate use of inequality symbols to compare numbers?
2. Can students round numbers to the nearest 10 and 100.
3. Can students use different strategies to solve addition and subtraction problems?
4. Can students add and subtract numbers of 3 digits or less?
5. Can students make informed choices about the appropriate use of problem-solving strategies?
6. Can students determine when and how to use computational strategies?
Grade 3 Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS)
Grade Level ExpectationsGLE # / GLE Text and Benchmarks
Number and Number Relations
2. / Read, write, compare, and order whole numbers through 9999 using symbols (i.e., <, =, >) and models (N-1-E) (N-3-E)
11. / Add and subtract numbers of 3 digits or less (N-6-E) (N-7-E)
13. / Determine when and how to estimate, and when and how to use mental math, calculators, or paper/pencil strategies to solve addition and subtraction problems (N-8-E) (N-9-E)
CCSS for Mathematical Content
CCSS # / CCSS Text
Numbers and Operations in Base Ten
3.NBT.1 / Use place value understanding to round whole numbers to the nearest 10 or 100.
ELA CCSS
CCSS # / CCSS Text
Writing Standards
W.3.2 / Write informative/explanatory texts to examine a topic and convey ideas and information clearly.
Sample Activities
Activity 1: The Largest Number Game (GLE: 2)
Materials List: deck of cards with face cards removed, base-10 blocks or paper models of base-10 blocks, number cubes, paper, pencil, Place Value Chart BLM
Give pairs of students a deck of cards with face cards removed and have each student draw four cards. Have each student make a 4-digit number with his/her cards. Have students create and model the largest four-digit number with base-10 materials (i.e., base-10 blocks or paper models of base-10 blocks). Have students compare their numbers by comparing the base-10 block models. Have students read their numbers aloud to their partners. The student with the largest number keeps all eight cards played. Continue until all cards are used.
Variations:
(1) Use number words instead of numerals on the cards.
(2) Students can use place value charts with thousands, hundreds, tens, and ones labeled at the top of four columns to demonstrate their numbers before using the base-10 blocks to model the number.
(3) On a paper with the students’ names printed across the top, have each student in the pair write his/her number in standard form under his/her name. In the middle, one student writes the correct sign for comparing the numbers (< or >).
(4) Instead of using the highest number, use the lowest.
(5) Use number cubes instead of cards.
Activity 2: Somewhere Between (GLE: 2; CCSS: W.3.2)
Materials List: paper, pencil
Place students in groups of three. Have Student 1 write a four-digit number, Student 2 write a second four-digit number that is not consecutive to the number written by Student 1, and Student 3 write a number that is between the first two numbers. Have each student then write three statements using the numbers created and each of the inequality symbols (<, >, and ) at least once.
After the students have done this, have them create a text chain (view literacy strategy descriptions) using four-digit numbers and the words “greater than,” “less than,” and “unequal.” Students can share their stories at the conclusion of the lesson as a review of <, >, and ≠. An example of a story chain might be as follows:
Student 1: I have 2,453 buttons. My number of buttons is greater than the number of buttons that (next student’s name goes here) has.
Student 2: I have 987 buttons. I have fewer buttons than (next student’s name goes here) has.
Student 3: I have 3,643 buttons. This number is unequal to the number of buttons that (student’s name goes here) has.
Activity 3: Four-Digit < and > (GLE: 2)
Materials List: paper, pencil
Have students work in pairs. Ask Student 1 to write a four-digit number and the symbol < or > with Student 2 completing the number sentence with a four-digit number. Continue the activity having the students alternate roles.
Extension: Have Student 1 write a four-digit number and the equal sign (=). Student 2 then completes the number sentence by writing the number in words or in expanded form.
Activity 4: Addition Properties (GLE: 11; CCSS: W.3.2)
Materials list: paper, pencil, board, counters
Tell students that they will be using the split-page notetaking (view literacy strategy descriptions) method. When using this method, a line is drawn down an 8 ½” by 11” lined paper about 2 ½” from the left margin. Big ideas or key words are written on the left side of the paper and facts or definitions and examples are written to the right. The page can be used as a study tool by folding it on the line or covering up one side of the page and using the information in the other column to recall the covered information. There is an example of what this looks like at the end of this activity.
Tell the students the following: If you know the fact 8 + 7, you also know the fact 7 + 8. Ask them to prove that this is true. They can use counters to prove that it is true. Tell students that there is a property in mathematics which states that changing the order in which numbers are added does not change the sum. Have students write “commutative property” on the left side of the paper, then describe what the property means and give an example on the right side of the paper.
Ask the students to add the following 6 + 8 + 2. Ask if anyone added 8 + 2 first. Tell them that the associative property of addition states that changing the way the addends are grouped does not change the sum. So they could add 6 + 8 first and get 14 and then add 2 to 14 to get 16. Or they could add 8 + 2 to get 10 and then add 6 to get 16. This means that (6 + 8) + 2 = 6 + (8 + 2). Have students write “associative property” on the left side of the paper, and then describe what the property means and give an example on the right side of the paper.
Ask the students how many students are in class today (e.g., 24 students). Say that no new students will join the class at any time this hour and no students will leave the class at any time this hour. Ask how many students will be in the class at the end of the hour (24). The number of students at the end of the hour could be represented by 24 + 0 = 24 or by 24 – 0 = 24. Tell students that this shows the addition property of zero and the subtraction property of zero. Have students write “zero property of addition” on the left side of the paper, and then describe what the property means and give an example on the right side of the paper. Have them do the same for the subtraction property of zero. Ask students what happens when zero is subtracted from a number. (The difference is that number.) Also, ask students what happens when a number is subtracted from itself. (The difference is always zero). Give pairs counters and have them demonstrate problems following these rules such as 7 – 0 = 0 or 7 – 7 = 0. Have students give their partners a problem which demonstrates one of these rules and have the partners state the rule. After the first student has modeled a rule, reverse the partners’ roles.
Have pairs of students switch papers and check their examples. These notes can then be used to study and refer to as students prepare for tests or quizzes.
Example of split-page notetaking:
Commutative Property Changing the order in which numbers are added does not
change the sum. 12 + 3 = 15 3 + 12 = 15
Activity 5: The Relationship Between Addition/Subtraction (GLE: 11)
Materials List: index cards, pencil
Explain to students that a fact family is a group of number sentences that use the same numbers. Tell students that fact families can be used to show how addition and subtraction are related. Model a few fact families. Have students create index cards of fact families after giving them three related numbers. Continue giving students 3 related numbers and have them write fact families. Give 3 numbers that are not related such as 6, 7, 14. See if students notice that these numbers do not make a fact family.
Example of Fact Family Cards
These cards can be collected and used in a center to study facts for those students that are not proficient.
Activity 6: Composing New Units (GLE: 11; CCSS: W.3.2)
Materials List: base-10 blocks, learning logs, pencils, Composing New Units BLM
Teacher Note: The Common Core State Standards use the terms composing and decomposing rather than regrouping. This is because ten ones are needed to compose or create a ten-unit and a ten-unit can be broken apart or decomposed into ten ones. It is important to begin to use this terminology during the transition and also to use strategies which help students understand that the standard algorithm is based on adding and subtracting of place values. Standard algorithms for addition and subtraction are not expected to be mastered until fourth grade. In this course, it is important that students are able to add and subtract numbers of 3-digits or less, but they should be allowed to use place value strategies when doing so.
Have students sit in a circle and ask a volunteer to use base-10 blocks to model the number 156. Have another volunteer model the number 128. Have students put the hundred blocks together. Ask how many hundreds they have now (2). Write the problem on the board and indicate to students that they are going to record what they see in the base-10 blocks in a different way, then write 200 as the first partial sum as shown below.
156
+ 128
------
200 (1 hundred + 1 hundred = 2 hundred or 200)
Next, have volunteers put the ten rods together. Ask how many tens they have (7). Ask students how to write 7 tens as a number (70). Then record the sum of the tens as shown below:
156
+ 128
------
200 ( 1 hundred + 1 hundred = 2 hundred or 200)
70 (5 tens + 2 tens is 7 tens or 70)
Next, have volunteers put the ones blocks together. Ask how many ones they have altogether and where they should write this in the problem. (14, write it under the 70). Ask students what 14 means in terms of tens and ones (1 ten and 4 ones) and lead them to understand that the 1 must be placed under other numbers which are in the tens place, and that the 4 must be entered in the ones place.
156
+ 128
------
200 (1 hundred + 1 hundred = 2 hundred or 200)
70 (5 tens + 2 tens is 7 tens or 70)
14 (8 ones + 6 ones is 14 ones which is 1 ten and 4 ones)
Show students how to complete the problem by adding the partial sums.
156
+ 128
------
200 (1 hundred + 1 hundred is 2 hundred or 200)
70 (5 tens + 2 tens is 7 tens or 70)
14 (8 ones + 6 ones is 14 ones which is 1 ten and 4 ones)
284 (2 hundred, 7 tens + 1 ten or 8 tens, 4 ones)
Point out to students that the ones were added to ones, tens were added to tens, and hundreds were added to hundreds. Make sure that students understand that this is done because addition is based on finding the sum of the numbers in each place value in the same way that the base-10 blocks were grouped together to determine how many ones, tens, and hundreds were in the sum.
Note that the process of finding partial sums by adding numbers in the same place starts on the left and goes to the right. This is natural for students as they read from left to write; however, some students may be comfortable in using the right-to-left process which is more aligned with the process used for the standard addition algorithm.
156