2012-13 and 2013-2014 Transitional Comprehensive Curriculum
Grade 5
Mathematics
Unit 2: Multiplication and Division of Whole Numbers and Decimals
Time Frame: Approximately six weeks
Unit Description
Unit 2 provides the closure to whole number work and provides the opportunity for students to begin the process of becoming computationally fluent in the whole number system by the end of the school year. Computational fluency is the level of skill reached when a person is able to execute an algorithm or procedure efficiently and correctly without assistance. This unit will also introduce multiplying and dividing by decimals. Work with whole numbers and decimals should be integrated in each subsequent unit.
Student Understandings
Students solidify their comprehension of whole numbers and the operations of multiplication and division. They understand numbers, ways of representing numbers, relationships among numbers, patterns in numbers; they can compute fluently, and can make reasonable estimates. Students multiply and divide numbers using models, diagrams, and their understanding of place value.
Guiding Questions
1. Can students determine the steps and operations to use to solve a problem without assistance?
2. Can students use mental mathematics and estimation strategies in checking the reasonableness of computations?
3. Can students work proficiently to multiply and divide whole numbers?
4. Can students solve simple equations and inequalities involving whole numbers?
5. Can students identify a simple rule for a sequence pattern problem and find missing elements?
6. Can students multiply and divide decimals using models, diagrams, and their understanding of place value.
Unit 2 Grade Level Expectations (GLEs) and Common Core State Standards (CCSS)
Grade-Level ExpectationsGLE # / GLE Text and Benchmarks
Number and Number Relations
7. / Select, sequence, and use appropriate operations to solve multi-step word problems with whole numbers (N-5-M) (N-4-M)
8. / Use the whole number system (e.g., computational fluency, place value, etc.) to solve problems in real-life and other content areas (N-5-M)
9. / Use mental math and estimation strategies to predict the results of computations (i.e., whole numbers, additions and subtraction of fractions) and to test the reasonableness of solutions. (N-6-M) (N-2-M)
Algebra
14. / Find solutions to one-step inequalities and identify positive solutions on a number line (A-2-M) (A-3-M)
Measurement
23. / Convert between units of measurement for length, weight, and time, in U. S. and metric, within the same system (M-5-M)
CCSS for Mathematical Content
CCSS# / CCSS Text
Operations and Algebraic Thinking
5.OA.1 / Use parentheses, brackets, or braces in numerical expressions, and evaluate expression with these symbols.
Number and Operations in Base Ten
5.NBT.5 / Fluently multiply multi-digit whole numbers using the standard algorithm
5.NBT.7 / Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
ELA CCSS
CCSS# / CCSS Text
Writing Standards
W.5.2a / Write informative/explanatory texts to examine a topic and convey ideas and information clearly.
a. Introduce a topic clearly, provide a general observation and focus, and group related information logically; include formatting (e.g., headings), illustrations, and multimedia when useful to aiding comprehension.
Speaking and Listening Standards
SL.5.1c / Engage effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grade 5 topics and texts, building on others’ ideas and expressing their own clearly.
c. Pose and respond to specific questions by making comments that contribute to the discussion and elaborate on the remarks of others.
SL.5.4 / Report on a topic or text or present an opinion, sequencing ideas logically and using appropriate facts and relevant, descriptive details to support main ideas or themes; speak clearly at an understandable pace.
Language Standards
L.5.6 / Acquire and use accurately grade-appropriate general academic and domain-specific words and phrases, including those that signal contrast, addition, and other logical relationships.
Sample Activities
Activity 1: Multiplication Properties (CCSS: L.5.6)
Materials List: Multiplication Properties BLM, pencils
Before beginning the multiplication activities, have students complete a vocabulary self- awareness chart (view literacy strategy descriptions). Vocabulary Self-awareness is a strategy that allows the student to gauge their prior knowledge of the terminology that will be used in understanding the concept. Vocabulary Self-awareness highlights the students’ understanding of what is already known as well as what is still needed to learn.
Provide students with the Multiplication Properties BLM. Do not give students definitions or examples at this point.
Word / + / Ö / – / Example / DefinitionCommutative Property
Associative Property
Distributive Property
Identity Property
Zero Property
Ask students to rate their understanding of each word with either a “+” (understands well), a “Ö” (some understanding), or a “–” (don’t know). During, and after completing any multiplication activities, such as activities 2, 3, 4 and 12, students should return to the chart and fill in examples and definitions in their own words. The goal is to have all plus signs at the end of the activities with appropriate examples and accurate definitions.
Activity 2: Properties and Mental Math (GLEs: 8, 9; CCSS: 5.NBT.5, W.5.2a)
Materials List: Grid Paper BLM, pencils, math learning logs
The multiplication properties can help students use mental math to multiply. Give students problems such as 4 × 18 × 25. Help them understand that using the Commutative Property to change the order of 18 and 25 makes the problem easier.
4 × 25 = 100 and 100 × 18 = 1800
Give students problems that the Associative Property would make easier, such as 34 × 4 × 5. Grouping the 4 × 5, and multiplying it first to get 20, leaves the easier problem of 34 × 20, or 680.
Give examples of the identity and zero properties of multiplication, such as 310 × 1 = 310 and 562 × 0 = 0.
The Distributive Property is extremely useful when multiplying mentally. Distribute the Grid Paper BLM. Have students draw a rectangle 6 units long and 13 units wide. Since 13 is the same as 10 + 3, have them draw a line to show 2 rectangles, one that is 6 × 10 and one that is 6 × 3.
This is an illustration of the Distributive Property.
10 + 3
13
6
6 × 13 = 6 × (10 + 3)
= (6 × 10) + (6 × 3)
= 60 + 18 = 78
Continue with other examples using the grid paper and then have students use the distributive property to multiply mentally. In their math learning logs (view literacy strategy descriptions), have students explain how they could use the distributive property to solve the problem 8 ´ 49. Allow students time to compare their explanations with a partner and make changes, if needed.
Activity 3: Multiplication With Regrouping (CCSS: 5.NBT.5, SL.5.1c, SL.5.4)
Materials List: base-ten blocks, pencils, math learning logs
Give base-ten manipulatives to each student. Tell students that these tools can be used to find the product of two factors. Demonstrate by using the problem: 24 ´ 3 = 72.
Model how to decompose the number 24 into 20 and 4. Display 3 groups of 2 rods and 3 groups of 4 units to show 3 ´ 20 plus 3 ´ 4. Ask students what 3 groups of 4 ones equals. (12 ones) Demonstrate how to regroup 10 of the 12 ones as one rod and 2 ones. (See figure shown below.) Add the one ten that was regrouped to the tens place. 3 groups of 2 tens equals 6 tens. Add the regrouped 1 ten to get 7 tens. Add the 2 units to get 72. Pair students together, combining base-ten blocks.
Provide problems such as 15 ´ 3, 53 ´ 2, 12 ´ 5, etc. for practice.
Model how to decompose a 3 digit × 1 digit number such as 212 × 2. Ask students to decompose the number 212 into hundreds, tens, and ones (2 hundreds + 1 ten + 2 ones). Have students make 2 groups of 2 hundreds, 1 ten, and 2 ones. They should see that the product is 424. Continue modeling other problems such as 150 × 5, 324 × 2, and 225 × 3.
Tell the students they will multiply a four digit number by a 1 digit number without the use of the blocks. Display the multiplication problem 4,205 × 2. Foster understanding about what is being multiplied by asking the students to decompose the number 4,205. (4,000 + 200 + 5) Have the students visualize each value doubling or being multiplied by 2. Ask the students to find the answer (8,000 + 400 + 10 = 8410). Work with students to solve the following problems:
3,219 × 3 = (3000 + 200 + 10 + 9) × 3 = 6,000 + 600 + 30 + 27 = 9,657)
5,291 × 9 = (5,000 + 200 + 90 + 1) × 9 = 45,000 + 1,800 + 810 + 9 = 47,619
1,025 × 4 = (1,000 + 20 + 5) × 4 = 4,000 + 80 + 20 = 4,100
6,689 × 6 = (6,000 + 600 + 80 + 9) × 6 = 36,000 + 3600 + 480 + 54 = 40,134
Clarify any misconceptions.
Have students create their own multiplication problems (more base-ten blocks may be needed per pairs). Call on students randomly to demonstrate their problems. In their math learning logs (view literacy strategy descriptions), have students write about the importance of regrouping units when there are more than 9 units. Allow time for students to compare their log entries with a partner and make changes, if needed.
Activity 4: Area Model of Multiplication (CCSS: 5.NBT.5, SL.5.1c, SL.5.4)
Materials List: Area Model of Multiplication BLM, pencils
The area model of multiplication helps students to organize the parts of a multi-digit multiplication problem and to understand the process of multiplication. A process guide (view literacy strategy descriptions) accompanies this activity and will be used to teach the area model of multiplication. A process guide is a strategy that scaffolds students’ comprehension within unique formats. Student thinking and involvement during the problem-solving process is enhanced as students focus on important ideas and concepts about the content area topic. Provide the students with the Area Model of Multiplication BLM and have them review the guide. Group the students into small groups. Explain the guide’s features (explore, explain, understand, apply, and reason) and tell them that they will use the process guide to work through the area model of multiplication. Tell them that this process guide will help them understand the parts of multi-digit multiplication problems.
Guide the students through the process by helping them see how the area model of multiplication is used to solve 37 ´ 23. Allow for discussion and listen as they explain how breaking the rectangle 37 ´ 23 into smaller rectangular sections of the whole leads to multiplying the tens and ones separately. Guide students in understanding this idea. Facilitate their completion of the guide, providing feedback and additional explanation as needed. The following are ideas for students to understand through the process guide:
· With the area model of multiplication, students should multiply the top row (tens then ones) and then the bottom row (tens then ones).
· Students may write the partial products in any order and may write the factors in different orders on the rectangle. (For example, in the problem 49 ´ 34, the 49 can be written on the left instead of the top.)
· Students can use what they know about zeros patterns in products to find partial products. When multiplying a number by 10, the number will shift one place to the left (number ´ 10). When multiplying a number by 100, the number will shift two places to the left (number ´ 100 or number ´10 twice).
· Students can also discuss how to decide the number of zeros in the partial product. When multiplying 40 ´ 30 to find one of the partial products in 49 ´ 34, the student can decide how many zeros belong in the partial product. One process is to multiply the two factors without the zeros and then add as many zeros as previously ignored.
Activity 5: Meanings of Division (GLE: 8; CCSS: 5.NBT.5)
Materials List: counters, pencils, paper
Have students work in pairs or small groups. Each pair will need about 40 counters. This activity focuses on the inverse relationship of multiplication and division, and strategies used in interpreting division.
Tell students the following:
When you think of multiplication as equal groups of an amount, you can write the following equation:
number of groups × number in each group = whole
If you know the whole amount and the number of groups, you can think of division as a way of equally sharing the whole amount among the groups.
whole ¸ number of groups = number in each group
If you know the number in each group and the whole amount, you can think of division as repeatedly subtracting to find the number of groups.
whole ¸ number in each group = number of groups
Understanding these strategies will help students later when solving more difficult division problems and with division of fractions. Give students many examples to model division with counters such as the following:
There are 5 groups of 4 students in the class. How many students are in the class? (20 students)
There are 20 students in the class. If they are separated into 10 groups, how many are in each group? (2) Explain how you interpreted (understood) the division problem? (The student may answer by sharing the amount or may say that they found the number in each group.)
There are 20 students in the class. If I send 4 students to each learning station, how many learning stations do I need? (5) Explain how you interpreted (understood) the division problem?
Teacher Note: (The student may answer by using repeated subtraction or may say that they found the number of groups.)
Continue with additional problems.
Activity 6: Order of Operations (GLE: 8; CCSS: 5.OA.1, 5.NBT.5, W.5.2a, SL.5.1c)