Ideas taken from Zeroing in on Number and Operations by Anne Collins & Linda Dacey
Grades 5-6
Topic #1: Millions and billions Billion looks a lot like a million, therefore students think the size is similar as well. This is a big misconception. Ten million and 100 million are not mentioned very often. Teachers need to say a billion is 1000 million.
Handout A-1
Topic #2: Place value The greatest challenge facing teachers is helping students develop facility with and knowing the importance of place value rather than just memorizing the names of the places for each digit.
Handouts A-2, A-3, A-4
Topic #3: Fact practice There is a difference between practice and drill. We need to stay away from “drill and kill.” Practice means we are doing math that we have recently learned in an attempt to achieve proficiency. Drill is activity designed to promote automaticity and speed in recall. Drill can cause anxiety in kids. Regular practice is essential.
Handout A-6
Topic #4: Finding Primes By definition, a prime number is a positive integer that has two and only two unique positive factors. The two most pervasive misconceptions are that one is a prime number and all primes are odd. One is special in that it has only one unique factor.
Handout A-7
Topic #8: Division algorithm The least understood of algorithms. It works left to right, rather than right to left. For 72 ÷ 4, we say, “how many fours are there in 72?” Not four goes into
(gozinta) seventy-two.

Example #1: The department secretary has 563 stamps to give to 4 workers. She has 5 sheets of 100 stamps, 6 strips of 10 stamps, and 3 single stamps. How many stamps should each worker get?
Solution: Solve by using the traditional algorithm. Then, model this problem using the base 10 blocks. Spread 4 plates across the table. Each worker gets 100 stamps right away. This leaves one sheet of 100 stamps, 6 strips of 10 stamps, and 3 single stamps. Compare back and forth with the model and the algorithm. Eventually, decide that the secretary keeps the remaing 3 single stamps.
Handout A-10
Topic #9: Greatest Common Factor (GCF) and Least Common Multiple (LCM).
Use Venn diagrams and prime factors to determine GCF and LCM. Find the prime factors of sixteen and twenty-four. 16→ 2, 2, 2, 2 24→ 2, 2, 2, 3 Intersect the two diagrams. They have three twos in common. Therefore the GCF = 2∙2∙2 = 8. How would you find the LCM?
Handout A-11, A-12
Topic #10: Working with remainders Research provides evidence that many students do not know how to interpret remainders. You must have computation with context. “Naked number” is using computation without context!
Example #1: The teacher hands out 25 movie passes to 37 students. Each student gets one.
Then the teacher hands out the remaining 12 passes to the first 12 students. The 13 th student complains that he only got one. After discussion the students decide to save the extra passes until the teacher has enough for everyone to be ‘fair.” Each student only gets one pass.
Handout A-14
Topic #11: Estimating Quotients Many students mistakenly believe that estimation means you complete the computation and then round the answer up or down. To estimate 4321 ÷ 73 it is not good to take 4000 ÷ 70. Round the dividend to the nearest factor or 4200 ÷ 70 = 60.
Handout A-15

Topic #12: Order of operation PEMDAS – Please Excuse My Dear Aunt Sally – Parentheses,
Exponents, Multiplication, Division, Addition, Subtraction – Used as a guide to know what to do first. This can promote a misconception with students as they may neglect doing multiplication and division from left to right.
Handouts A-16, A-17
Topic #13: Fractions on the number line Fractions are used in 3 distinct ways: 1) as numbers,
2) as ratios and 3) as division. Teachers should select the manipulatives they’ll use in the teaching and learning of fractions according to the meaning of the fractions being used.
Fractions as number representations answer the question How Much? Ratio representations answer the questions How Many?
Handouts A-18, A-19
Topic #14: Comparison of fractions One of the greatest errors you can make with students is comparing fractions by changing them to decimals before students have mastery of fractions at a conceptual level. The National Math Advisory Panel (2008) recommendation is to encompass reasoning with fractions as well as computing with fractions. A lack of conceptual understanding of fractions will interfere with a students success in algebra.
Example: Circle the larger fraction. Tell how you decided.
1) 7/11 or 9/11 _____________________________________
2) 14/1000 or 14/100 ______________________________________
3) 5/13 or 7/13 ________________________________________
4) 5/8 or 8/11 __________________________________________
5) 19/9 or 15/8 ________________________________________
6) 7/8 or 11/12 _____________________________________________
7) 2/5 or 4/7 ______________________________________________
8) 20/17 or 17/20 ______________________________________________
Handout A-19

Topic #15: Adding and subtracting fractions with pattern blocks Current instruction of fractions is not serving many students well. They think (because of whole number thinking) that 2/3 + 5/6 = 7/9. Student drawn circles may cause misconceptions because the parts are many times not equivalent. Use pattern blocks to lessen these misconceptions.
Handout A-20, A-21, A-22
Topic #16: Modeling multiplication of fractions Without models and representations that make sense to students, they are often confused when multiplication of fractions yields a product that is less than either one of the factors. The algorithm is easy, but masks lack of understanding of the concept. Fraction patterns: Hexagon is one, then the trapezoid is onehalf, the larger rhombus is one-third, and the triangle is one-sixth.
Example #1: Ms. Johs is making puppets. She needs ½ of a yard of brown felt to make one puppet. How many yards of brown felt does she need to make 6 puppets?
Solution: Model ½ of a yard with a trapezoid and then gather six trapezoid one for each puppet. Next, configure the six trapezoids to make 3 hexagons. Therefore, the total amount of cloth is 3 yards.
Handout A-20, A-23
Topic #17: Modeling division of fractions Your is not the reason why, just invert and multiply.
Why does inverting and multiply work? Most kids are surprised when: 2 ÷ ⅓ = 6. How can you get a bigger answer? The measurement model of division, which focuses on the number of groups when the size of the group is known, is the easiest model to represent and so that is the best place to begin.
Example: I have
½ cup of sugar and sweet bread requires
⅙
cup of sugar. How many loaves of sweet bread can I make? How many one-sixths are there in one-half?
Using pattern blocks it says how many triangles (
⅙
) are there in a trapezoid (
½
)? _______
Handout A-24

Topic #18: Multiplying fractions with arrays Too many students do not understand that when you multiply two numbers you get a rectangle. Multiplication is used as a quicker way to add in some cases, but not always!
Example #1: I have one pan of brownies. What is ½ pan of ½ pan of brownies?
Solution:
½∙½ = ¼
(algorithm) or Model this by taking a whole sheet of paper and fold it in half and then fold it in half again. This will result in
¼ of the paper.
Example #2: ⅔∙⅔ =
Solution: A whole rectangle is divided into thirds along the horizontal and vertical sides. You have ⅔ of the bottom and ⅔ of the right side. This totals 4 out of the 9 squares are shaded which equals 4/9.
Handout A-25
Topic #19: Division of fractions Handouts A-26, A-27
Topic #20: Problem solving with fractions Handouts A-29, A-30
Topic #21: Converting fractions to decimals Handout A-33
Topic #22: Equivalent values Handout A-34
Topic #23: Estimating with decimals Handout A-35
Topic #24: Adding and subtracting decimals There are numerous misconceptions about decimals: 1)Longer decimals are larger, 2)Longer decimals are smaller, 3)Putting a zero at the end of a decimal makes it 10 times larger, 4)Decimals are below zero or negative numbers.
Handout A-36

Topic #25: Multiplying decimals using arrays Make a 10 by 10 grid. Label the bottom horizontal grid tenths (0, .1, .2, .3, .4, .5, .6, .7, .8, .9). Do the same with the right hand vertical side. Multiply .1 x .1 = .01 or 1/100 of the whole grid.
Handout A-37, A-38
Topic #26: Dividing decimals Students must be proficient with division of whole numbers before they work on division of decimals.
Example: The teacher collected $10.35 for snacks. Each snack costs $.45. How many students paid for their snack? Use a multiplication menu to help solve this problem.
1 x .45 = .45
10 x .45 = 4.5
20 x .45 = 9.0
30 x .45 = 13.5 (This is more than $10.35, so there are less than 30 students who paid)
5 x .45 = one-half of 4.50 is 2.25
25 x 45 = 9.00 + 2.25 = 11.25 (Darn it, too big)
2 x .45 = .90
3 x .45 = 1.35
23 x .45 = 9.00 + 1.35 = 10.35 (Whew, that’s it!) 23 students have paid for their snack.
Handouts A-39, A-40
Topic #26: Rates and ratios Rates use the word per. 60 mph, 23 mi/gal, $.23 per orange. Unit rates are for one. Finding unit rates are essential for proportional thinking and algebra.
Handout A-41