U



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 Expectations |

|GLE # |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. |

| |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.|

| |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 |+ |( |– |Example |Definition |

|Commutative Property | | | | | |

|Associative Property | | | | | |

|Distributive Property | | | | | |

|Identity Property | | | | | |

|Zero Property | | | | | |

Ask students to rate their understanding of each word with either a “+” (understands well), a “(” (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

| |

|5 − [3 − 3] |

|5 − 0 |

|5 |

Inform students that math expressions can be even more complex than the problems previously experienced. Braces can be used to enclose sections of the expression that have brackets with parentheses inside.

Have the students evaluate the expression 6{3[12 – (4 + 6)] – 5}. The following shows the way the problem should be worked:

|6{3[12 – (4 + 6)] – 5} |

|6{3[12 – 10] – 5} |

|6{3[ 2} – 5} |

|6{6 – 5} |

|6{1} |

|6 |

Ask students to discuss with a partner how the problem was worked. Have students work the following problem in their math learning logs (view literacy strategy descriptions):

5 – [3 − (5 − 2) + 2]. The following shows the way the problem should be worked:

|5 – [3 − (5 − 2) + 2] |

|5 – [3 – 3 + 2] |

|5 – [0 + 2] |

|5 – [2] |

|3 |

Give each student a copy of the Parentheses, Brackets, and Braces BLM. Have students work with a partner to solve the math expressions. Assist pairs as needed. When all problems are complete, allow students to share their experiences in solving the expressions.

Activity 8: Estimation in Division (GLEs: 8, 9, 23; CCSS: W.5.2a)

Materials List: paper, pencils, math learning logs

The types of estimation used for division at this grade level are rounding and compatible numbers. Give students a problem and have them estimate using both of these methods. Use measurement units to practice measurement concepts and to put problems in context.

Example 1: If there are 8 ounces in one cup, about how many cups are in 62 ounces?

Rounding: 60 ÷ 8 = ?

This problem is still not very easy. You know it is more than 7 cups, but less than 8 cups. Since you used 60 rather than 62, you know that 7 is an underestimate.

Compatible numbers: 64 ÷ 8 = 8

This problem was easy to divide. Eight cups is an overestimate because you used 64 instead of 62.

Example 2: If there are 12 inches in one foot, about how many feet are in 34 inches.

Rounding: 30 ÷ 10 = 3 feet

Compatible numbers: 36 ÷ 12 = 3 feet

Both estimation methods give 3 feet.

Continue giving division problems using other measurements units. In their math learning logs (view literacy strategy descriptions), have students explain how they could estimate the number of feet in 43 inches. With a partner, let students share their explanations and make any modifications, if needed.

Activity 9: Division (GLEs: 7, 8, 9)

Materials List: base-ten blocks, Grid Paper BLM, pencils, paper

Have students work in groups and provide each group with base-ten blocks and grid paper. Tell students that 500 flyers were copied to advertise the school fair. If each person in the class takes an equal amount, how many flyers will each student get and how many will be left over? Use the number of students in the class. The example below uses 23. Allow students to use any of the materials they want to model the problem.

Observe how the students approach the problem. Did the students use estimation techniques such as rounding or compatible numbers? Did the students use repeated subtraction? Did they make a rectangular array with the blocks? Did they draw a rectangle on the grid paper?

Suppose there are 23 students in the class. Ask students questions such as these: Can you give each person a set of 100 flyers? (No) Could you give each person a set of 10 flyers? (Yes) How many sets of 10 could you give each person? (2 sets) How many sets of flyers did you give out altogether? (46 sets) How many are left? (40) Can you give each person some individual flyers? (Yes) How many will you give each person? (1) How many did you use? (23) How many are left over? (17) Consider allowing students to write the division problem the second way.

1

21 20

[pic] [pic] 21 R 17

46 460

40 40

23 23

17 17

Continue giving students other division problems. Have students write the word problems that will be used in the class. Encourage students to write some problems that involve multiple steps using any of the four operations.

Activity 10: Rectangle Sections Method of Division (GLE: 8; CCSS: 5.NBT.5, W.5.2a, SL.5.1c, SL.5.4)

Materials List: Rectangle Sections Method of Division BLM, pencils

The rectangle sections method is a visual method for division. It helps students visualize the division problem as a rectangle where the area is known as well as the length of one side. This method helps the students understand the long division process. A process guide (view literacy strategy descriptions), accompanies this activity and will be used to teach the rectangle sections method of division. Provide the students with the Rectangle Sections Method of Division BLM and have them survey the process guide. Group the students in 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 rectangle sections method of division. Tell them that this process guide will help them understand the missing factor in a division problem.

[pic]

Guide the students through the process by helping them see how the rectangle sections method of division is used to solve 4,823 ÷ 7 in this word problem. “An airplane travels the same distance every day. It travels 4,823 miles in a week. How far does the airplane travel each day?” Allow for discussion and listen as students explore how to work through the steps in the division problem. Facilitate their completion of the guide providing feedback and additional explanation as needed. The following are ideas students should understand from using the process guide:

• In division, the unknown factor is found.

• In long division, each place in the dividend is divided by the divisor.

• Students can use what they know about zeros patterns in products to find factors.

Activity 11: Multiplication and Division Text Chains (GLEs: 7, 8; CCSS: 5.NBT.5, W.5.2a)

Materials List: paper, pencils

Use math text chains (view literacy strategy descriptions) to practice any of the four operations with an emphasis on multiplication and division. The first student writes this possible opening sentence of the problem.

I have $512 to spend for the class.

The student passes the paper to the student sitting to the right. That student might write the next sentence as:

I want to buy 18 video games at $25 each.

The paper is passed again to the right. This student could then write the question for the story.

Do I have enough money to buy all 18?

The paper is now passed to the fourth student who must solve the problem and write the answer in a complete sentence. In this example, the last line of the text chain would be:

Answer: Yes, I have enough money because 20 × 25 is 500, and I have more than $500.

Students in the text chain groups should talk about the accuracy of the answer and the logic of the story problem. If necessary, revisions to the text chain should be made.

Activity 12: Which Method Would You Use? (GLEs: 7, 8, 9; CCSS: 5.NBT.5)

Materials List: Which Method? BLM, paper, pencils, math learning logs

Give students the Which Method? BLM. Using the numbers in the table below, have students write two problems for each of the following methods: mental math, calculators, paper and pencil. For the first problems for each method, use two numbers. For the second problem, use three numbers. Any operation can be used, but students must use every operation at least once. For example, a student might use mental math to solve the problem 4,381 ( 100.

|4,381 |38 |2,000 |

|100 |99 |8,296 |

|200 |635 |62 |

|19 |1 |4 |

In their math learning logs (view literacy strategy descriptions), have students explain their reasoning for one of the mental math problems. Partners can discuss mental math processes after making their log entries.

Have the students complete a SPAWN writing (view literacy strategy descriptions) activity that asks “Problem Solving” in a thought-provoking activity that relates to multiplication methods. SPAWN writing is a strategy that allows students to respond to prompts that stimulate students’ meaningful thinking about text they have read and topics they have learned. Ask students to answer this prompt: Through this activity you learned how to use mental math, calculators, and paper and pencil to solve problems. How would you describe situations when one method would be better to use than the others?

In groups of four, have students create situations where using mental math would be better to use than other methods and vice versa and write their ideas in their learning logs (view literacy strategy descriptions). Have students share their response with the class as they listen for accuracy and logic.

Activity 13: Does It Balance? (GLEs: 14)

Materials List: number or pan balances, objects to count, paper, pencils

Introduce students to the concepts involved in solving equations and inequalities. Have the students use the balances and similar objects (marbles, tiles, plastic counters) to create an equation or inequality. Have students write number sentences to show the reading of the balances: 8 > 6, 5 = 5, etc.

Using the balance, show students that multiplying or dividing both sides of the equation or inequality by the same amount (as long as you are multiplying or dividing by a positive number) does not change the relationship. For 5 = 5, if you multiply both sides by 2 (or double the amount), you get 10 = 10. For 8 > 6, if you divide both sides by 2 (or take ½ of each side), the left side of the inequality will still be greater than the right side. The inequality would now read 4 > 3.

Apply the ideas to larger numbers that cannot be done on the scales. If 482 = 482, does the equality change if you divide both sides by 2? (No, 241 = 241.) If 45 < 54, does the inequality change if you multiply by 3? (No, 135 < 162.)

Activity 14: One Grain of Rice (GLEs: 8; CCSS: 5.NBT.5, W.5.2a, SL.5.1c)

Materials List: One Grain of Rice: A Mathematical Folktale (optional), calculators, paper, pencils, math learning logs

Read the book, One Grain of Rice: A Mathematical Folktale, to the students. When you begin reading, have students predict the number of grains of rice on the 30th day. In their math learning logs (view literacy strategy descriptions), have students explain why they predicted their amount. As the book is read, write the amount of rice for each day as a sequence. 1, 2, 4, 8, 16, … Allow students to change their predictions after day 5, day 10, day 15, and day 20. Each time students change their predictions, they should explain their reasoning in their math learning logs. Ask students to describe any patterns they see. (Each day the amount doubles.) If the book is not available, just focus on the pattern in the sequence. Also show this pattern of growth on a calculator. Enter[pic]. (On some 4-function calculators, enter[pic].) Because the numbers get large so quickly, ask questions about place value, estimation, and comparing with large numbers.

Give students the following problem: You are offered a job, which lasts for 7 weeks. You get to choose your salary.

• Either, you get paid $100 for the first day, $200 for the second day, $300 for the third day, etc. Each day you are paid $100 more than the day before.

• Or, you get paid 1 cent for the first day, 2 cents for the second day, 4 cents for the third day, etc. Each day you are paid double what you were paid the day before.

Have students, in a group of 4, decide which job offer would be the best economical choice between the two given. Ask them to demonstrate their understanding of patterns in multiplication by two by completing a RAFT writing (view literacy strategy descriptions) assignment. This form of writing gives students the freedom to project themselves into unique roles and to look at content from unique perspectives. From these roles and perspectives, RAFT writing can be used to describe a point of view, envision a potential job or assignment, or solve a problem. It is the kind of writing that when crafted appropriately, should be creative and informative.

R – Role (role of the writer – potential employee)

A – Audience (to whom the RAFT is being written – human resource manager)

F – Form (the form the writing will take – a letter)

T – Topic (the subject of the focused writing – to formally state which salary offer you are willing to take and why.

The groups should share their writing with the class. Students should listen for accuracy and logic in each RAFT.

2013-2014

Activity 15: Decimal Patterns (CCSS: 5.NBT.7)

Materials List: learning log, pencils, calculator, Decimal Patterns Chart BLM

Have students work in pairs to complete the first portion of the activity. Give students a copy of the Decimal Patterns Chart BLM. The student should see that the answers are the same when they multiply by 0.1 and divide by 10.

|Number: |To multiply by: |Answer: |Number: |Divide by: |Answer: |

|10 |× 0.1 |1 |10 |÷ 10 |1 |

|9 |× 0.1 |0.9 |9 |÷ 10 |0.9 |

|8 |× 0.1 |0.8 |8 |÷ 10 |0.8 |

|7 |× 0.1 |0.7 |7 |÷ 10 |0.7 |

|6 |× 0.1 |0.6 |6 |÷ 10 |0.6 |

|5 |× 0.1 |0.5 |5 |÷ 10 |0.5 |

|4 |× 0.1 |0.4 |4 |÷ 10 |0.4 |

|3 |× 0.1 |0.3 |3 |÷ 10 |0.3 |

|2 |× 0.1 |0.2 |2 |÷ 10 |0.2 |

|1 |× 0.1 |0.1 |1 |÷ 10 |0.1 |

Have students discuss their answers with their partners to check for accuracy. Have students predict what would happen to the products if the numbers were multiplied by 0.01 or divided by 100. Have the students calculate the numbers on the remaining chart. Check that their completed charts are correct before moving on. Use as many numbers as needed to foster understanding of the patterns shown.

|Number: |To multiply by: |Answer: |Number: |Divide by: |Answer: |

|10 |× 0.01 |0.10 |10 |÷ 100 |0.10 |

|9 |× 0.01 |0.09 |9 |÷ 100 |0.09 |

|8 |× 0.01 |0.08 |8 |÷ 100 |0.08 |

|7 |× 0.01 |0.07 |7 |÷ 100 |0.07 |

|6 |× 0.01 |0.06 |6 |÷ 100 |0.06 |

|5 |× 0.01 |0.05 |5 |÷ 100 |0.05 |

|4 |× 0.01 |0.04 |4 |÷ 100 |0.04 |

|3 |× 0.01 |0.03 |3 |÷ 100 |0.03 |

|2 |× 0.01 |0.02 |2 |÷ 100 |0.02 |

|1 |× 0.01 |0.01 |1 |÷ 100 |0.01 |

Have students discuss their charts with their partners and explain any patterns or relationships they see.

Teacher Note: The student should see that when multiplying by 0.1, each number moves 1 place to the right as 1/10th of the quantity is taken. This is also true for dividing a number by 10.When multiplying by 0.01 or dividing by 100 1/100th of the quantity is taken; therefore, the number moves 2 places to the right. This is the same as taking 1/10th of the quantity and then 1/10th again.

Have the students complete a SPAWN writing (view literacy strategy descriptions) activity that asks “Problem Solving” in a thought-provoking activity that relates to multiplication methods. SPAWN writing is a strategy that allows students to respond to prompts that stimulate students’ meaningful thinking about text they have read and topics they have learned. Ask students to answer this prompt: Today we learned how the multiplication of decimals is similar to dividing by multiples of ten. Using the knowledge you gained today, answer the following questions: If given a choice of a money amount, which would you choose: 0.1 of $8; 0.01 of $24; (The best choice would be 0.1 of $8 which is 0.8 or 0.80 or 80 cents. The other choice is 0.24). Once your choice is made, tell why your choice is the best choice. In groups of four, have students discuss and choose the best answer and write their individual ideas in their math learning logs (view literacy strategy descriptions). Students should share their response with the class, as they listen for accuracy and logic.

2013-2014

Activity 16: Modeling Multiplication with Decimals (CCSS: 5.NBT.7, W.5.2a)

Materials List: Hundreds Grid BLM or grid paper, colored pencils or crayons, pencils, math learning logs

Give each student a copy of the Hundreds Grid BLM. Remind students that the grid represents 1 whole unit, that one column or row is 1/10 or 0.1, and that one square is 1/100 or 0.01 of the whole. Tell the students that they will model 0.8 × 0.4 using the hundreds grid. Model shading 0.8 of the grid by shading in 8 yellow columns vertically as shown below. Have students create the same model on their grid. Remind the student that 0.8 is equivalent to 0.80. Model shading 0.4, with a blue color. Start in the top left square and shade in 4 rows horizontally. Ask the students to explain what has happened when the 8 columns overlapped with the 4 rows. Students may state that green squares were made. This revelation will help the students understand that 0.8 of 0.4 will result in the 32 green shaded squares. Inform the student that the blended colored squares are the product. Have the students count the squares to find the product (0.32).

On this drawing and the next one, light grey was used to shade the columns, medium grey was used to shade the rows, and the resulting product is shown in dark grey.

| | |

| | |

Model how to multiply the decimals in each square to find the partial products. When multiplying, work the problem without the decimals. Show the students how to put the decimal point back into the answer by counting the total amount of numbers after the decimal point in the original expression, and moving the decimal point that amount of places to the left in the answer. The answers are shown above. Have the students add all the partial products to get the product (0.1728). Next, model how to use the area model of multiplication to multiply whole numbers and decimals. Have the students consider the expression 4.6 × 0.22. Have the students create another area model in their math learning logs (view literacy strategy descriptions). Model how to multiply the decimals in each square to find the partial products as shown below.

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| | |

Have the students add all the partial products to get the product (1.012). Ask students to solve the expressions shown on their Moving Methods of Multiplication BLM. Give assistance as needed. Have the students share their answers with partners.

2013-2014

Activity 18: Modeling Division (CCSS: 5.NBT.7, W.5.2a)

Materials List: Hundreds Grid BLM or grid paper, pencils, math learning logs

Start the activity by asking the students the following question, “How can models be used to divide decimals?” Give students time to discuss the answer. Display a hundreds grid. Review that each square of the model represents 0.01. Give the student a copy of the Hundreds Grid BLM. Have students work in pairs and use a hundreds grid to show 0.24 in various ways. Remind the students that 0.2 is equivalent to 0.20. Model how to divide 0.24 by 0.6 (0.24 ÷ 0.6) using the hundreds grid. Shade in 0.24 by shading 24 of the small squares in the model, making sure that the length is 0.6 as shown.

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The area of the model is 0.4 by 0.6 = 0.24. This model shows that 0.24 ÷ 0.6 = 0.4.

Model how to divide 0.2 ÷ 0.4. Since the model is in hundredths, display 0.2 as 0.20. Shade in a rectangle with an area of 0.20 and a length of 0.4. The missing factor is 0.5 as shown.

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Have students work in pairs using decimal models to show each of the following problems: 0.36 ÷ 0.6; 0.35 ÷ 0.5; 0.48 ÷ 0.8. The pairs should discuss solving the problems as they work. Students’ work may look like the models shown.

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Model how to divide 4 ÷ 0.2. Since the grid is in hundredths, four hundredths grids will be used and shaded. Have the student think of finding the number of groups that can be made by separating the four grids into 0.2 until all groups of 0.2 are made. They can model the equal groups by coloring each group of 0.2 a different color from the previous. The groups have been separated and counted below.

[pic][pic][pic][pic]

Students should see that although the whole number 4 was divided by the decimal number 0.2, the answer is a whole number (20). 20 equal groups of 0.2 were found in the whole number 4. Have the students work with their partner and model the following division problems: 4 ÷ .4 (10); 6.5 ÷ .5 (13); 7.2 ÷ 0.9 (8). After appropriate time is given to solving the problems, have students write a paragraph explaining how the models show the division of decimals in their math learning logs (view literacy strategy descriptions).

Sample Assessments

General Assessments

• Portfolio assessments could include the following:

o Anecdotal notes made during teacher observation.

o Any of the journal entries or one of the explanations from the specific activities

o Corrections to any of the missed items on the tests

• On any teacher-made written tests, the teacher could include at least one of the following.

o One problem that requires the use of manipulatives or drawings such as this: Using some type of base-ten manipulative or drawing, show 6 × 24.

o One question that requires the student to explain his/her reasoning such as this: How you solve the problem 56 × 3 mentally?

o One problem involving real-life such as this: Since numbers and mathematics are used all the time during the day, list two times that an exact answer is needed to answer a question, and two times that an estimate is all that is needed.

o One problem involving multiplying decimals where a model using a hundreds grid is used. A problem such as this could be used: 0.9 × 0.3, 0.3 × 0.3, 0.8 × 0.5, and 0.3 × 0.1.

• Journal entries will include the following:

o Explain how you would estimate the answer to the following problem: 135 × 19 = _____

o Mr. Mistake worked the following problem [pic]and got an answer of 2824. Explain why his answer is not reasonable, and tell the mistake he made.

o Explain in writing how to mentally find the product of 52 and 7.

o Make a diagram to show 0.4 × 0.5.

o Which product is greater 0.4 × 0.5 or 0.04 × 0.5? Explain your answer.

o Explain how to use a model drawing to divide 0.54 by 0.6 (0.54 ÷ 0.6).

Activity-Specific Assessments

• Activity 2: Ask students, “Which property or properties would you use to multiply 6 ( 90 ( 5 mentally?” (Possibly commutative property 6 ( 5 ( 90) “What is your answer?” (2700)

• Activity 12: Ask students which method, mental math, paper and pencil, or a calculator, would they use to solve the problem 8,296 ( 100? (The best response is mental math.) Have them explain their reasoning.

• Activity 16: Ask students to solve the following word problems:

o Joan brought 5 terrariums. Each holds 2.75 gallons of sand. How many gallons of sand will they hold altogether? (2.75 ( 5 =13.75 gallons)

o Keisha is making Fuzzy Furballs for her 6 friends. She needs 1.8 yards of fabric for each Furball. How much fabric will she need? (1.8 ( 6=12.8 yards)

Have students explain the multiplication method used to find each answer.

• Activity 18: Have students use decimal models to show each of the following problems: 0.40 ÷ 0.5; 0.56 ÷ 0.7; 0.49 ÷ 0.7. Have students explain one of their models.

-----------------------

0.32

0.4

0.4

0.24

0.8

0.8

0.35

0.5

0.7

0.35 ÷ 0.5 = 0.7

This model is incorrect.

0.36 ÷ 0.6 = 0.6

0.4 × 0.6 = 0.24

0.7 × 0.8 = 0.56

0.03 × 2 = 0.06

4 ( 6

4 ( 20

10 ( 6

10 ( 20

+

+

+

+

+

0.06

0.30

+

0.40 ( 0.30 = 0.1200

0.40

0.06 ( 0.40 = 0.0240

0.08

0.30 ( 0.08 = 0.0240

0.06 ( 0.08 = 0.0048

+

0.606

4

+

4 ( 0.20 = 0.8

0.204

0.60 ( 0.20 = 0.12

0.02

4 ( 0.02 = 0.08

0.60 ( 0.02 = 0.012

?

0.6

?

0.4

0.36 ÷ 0.6 = 0.6

0.35 ÷ 0.5 = 0.7

0.48 ÷ 0.8 = 0.6

0.36 ÷ 0.6 = 0.6

0.36 ÷ 0.6 = 0.6

9

15

14

13

12

10

7

6

5

2

1

3

4

8

16

17

18

19

20

11

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