Scaffolding/Constructing Task: Patterns-R-Us In this task, students are asked to identify, describe, and explain any patterns they notice when multiplying or dividing numbers by 1000, 100, 10, 0.1, and 0.01. STANDARDS FOR MATHEMATICAL CONTENT Understand the place value system. MCC5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. Perform operations with multi-digit whole numbers and with decimals to the hundredths. MCC5.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. STANDARDS FOR MATHEMATICAL PRACTICE SMP 1. Make sense of problems and persevere in solving them. SMP 2. Reason abstractly and quantitatively. SMP 3. Construct viable arguments and critique the reasoning of others. SMP 4. Model with mathematics. SMP 5. Use appropriate tools strategically. SMP 6. Attend to precision. SMP 7. Look for and make use of structure. SMP 8. Look for and express regularity in repeated reasoning. BACKGROUND KNOWLEDGE Students should develop an understanding that when a number is multiplied by a number less than 1, the product is less than the original number, and when a number is divided by a decimal number less than 1, the quotient will be greater than the dividend. This is important, yet difficult for students to understand because it is counterintuitive based on students’ previous experiences with multiplication and division. Because students have not yet learned how to multiply or divide decimals, calculators are recommended for this investigation. Students will be more likely to explore a variety of numbers and be able to recognize patterns more efficiently with the use of a calculator. Require students to record what they put into the calculator and the result. If students have learned how to multiply and divide
STANDARDS FOR MATHEMATICAL PRACTICE 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. BACKGROUND KNOWLEDGE If students use prior knowledge about whole number multiplication, they will realize that the products will have the exact same digits, but the decimal placement will change the value of the numbers. If students are methodical about their solution strategy, they will be able to find all of the possibilities systematically from 0.1 x 5 = 0.5 to 1.9 x 5 = 9.5. COMMON MISCONCEPTIONS Multiplication can increase or decrease a number. From previous work with computing whole numbers, students understand that the product of multiplication is greater than the factors. However, multiplication can have a reducing effect when multiplying a positive number by a decimal less than one or multiplying two decimal numbers together. We need to put the term multiplying into a context with which we can identify and which will then make the situation meaningful. Also using the terms times and groups of interchangeably can assist with the contextual understanding. ESSENTIAL QUESTIONS How can we efficiently solve multiplication and division problems with decimals? How can we multiply and divide decimals fluently? What strategies are effective for finding a missing factor or divisor? MATERIALS Calculators (optional) GROUPING Small group/Individual task TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION: Comments: This task will allow the teacher to see if students are able to use what they know about whole number multiplication and transfer it to decimal multiplication. Students will be able to come up with more solutions if they realize that whole numbers can be represented with
Three bottles of glue pencils Ten pens Five spiral notebooks Two backpacks Determine the cheapest for each item on your list of school supplies needed. From which store would you buy each school supply? How much money will you have to spend? How much of the $45.50 will you have left over? With the remaining money, you decide to buy each of your three siblings the same school supply item. What item would you purchase? Is one store better than the other? Why or why not? If you could choose only one store, which would it be? Explain your reasoning by writing an overall summary of your findings. See sale papers from two different stores shown below. CHEROKEE DISCOUNTS Calculator $6.99 Lined Paper 500 sheets 93¢ Folder 15¢ each Erasers – large 52¢ each Clipboard $2.15 Book Covers 4 for $2.00 Glue 2 oz. bottle 95¢ Backpack $8.97 Pencils 11¢each 10 pack -- $1.01 Scissors $1.75 Pens 29¢ each $3.56 – 15 pack Ruler $1.45 each Spiral Notebooks 99¢ each CANTON SUPPLIES Calculator $12.98 – regular Lined Paper 500 sheets Regular - 47¢ Folders 11¢ each Erasers – large 60¢ each Clipboard $1.99 Book Covers $1.50 Glue 2 oz. bottle 39¢ Backpack $8.37 Pencils 69¢ each 10 pack -- $2.02 Scissors $2.13 Pens 35¢ each $4.08 – 15 pack Ruler $1.14 each Spiral Notebooks 75¢ each