1. Ideas for Teaching Number Sense Including Facts and Number Operations

2. Math Facts: Strategies for Helping Students Who Struggle 1. Emphasize number sense patterns and relationships – number lines. How many different ways can you total six using these yellow cubes? Write the equation. two cubes and four cubes (2 + 4 = 6) two cubes and four cubes (2 + 4 = 6) 1 2 3 4 5 6 7 8 9 10 four cubes and two cubes (4 + 2 = 6) four cubes and two cubes (4 + 2 = 6) 1 2 3 4 5 6 7 8 9 10 How are the two representations similar? How are they different?

3. Math Facts: Strategies for Helping Students Who Struggle How many different ways can you total six using these yellow cubes? Write the equation. two cubes and four cubes (2 + 4 = 6) two cubes and four cubes (2 + 4 = 6) 1 2 3 4 5 6 7 8 9 10 four cubes and two cubes (4 + 2 = 6) four cubes and two cubes (4 + 2 = 6) 1 2 3 4 5 6 7 8 9 10 How are the two representations similar? How are they different? 1. Emphasize number sense patterns and relationships – number lines.

4. Math Facts: Strategies for Helping Students Who Struggle 1. Emphasize number sense patterns and relationships – number frames 10 line A number frame can be used to help students make sense of number patterns that represent addition and subtraction facts.

5. Math Facts: Strategies for Helping Students Who Struggle 1. Emphasize number sense patterns and relationships 10 line A number frame can be used to help students make sense of number patterns that represent addition facts by providing them with visual reference points. Working with students on a regular basis to identify different patterns can help them to create more meaningful cognitive anchors for recalling certain fact combinations

6. 2. Emphasize facts that are more easily remembered first – abstract recall Lower Stress Facts ExamplesWhy? Ones4 + 1; 9 + 1; 1 + 4; 1 + 9; 4 – 1; 9 – 1; 4 x 1; 9 x 1; 1 x 4; 1 x 9; 4 ÷ 1; 9 ÷ 1 Zeros4 + 0; 9 + 0; 0 + 4; 0 + 9; 4 – 0; 9 - 0 4 x 0; 9 x 0; 0 x 4; 0 x 9 4 ÷ 0; 9 ÷ 0; 0 ÷ 4; 0 ÷ 9 Doubles0 + 0; 1 + 1; 2 + 2; 4 + 4; 0 – 0; 1 – 1; 2 – 2; 4 – 4; 0 x 0; 1 x 1; 2 x 2; 4 x 4; 0 ÷ 0; 1 ÷ 1; 2 ÷ 2; 4 ÷ 4 Tens and Fives 10 + 2; 2 + 10; 5 + 2; 2 + 5; 10 – 2; 5 – 2; 10 x 2; 2 x 10; 5 x 2; 2 x 5; 10 ÷ 2; 20÷10; 10÷5; 20÷5

7. 3. Teach rules for multiplying facts (adapted from Mercer & Mercer, 2001) RuleExamples Order RuleThe order of numbers being multiplied does not affect the answer. (e.g., 2 x 4 = 4 x 2) Zero RuleAny number multiplied by 0 is zero (e.g., 2 x 0 = 0; 4 x 0 = 0) One RuleAny number (except zero) multiplied by 1 is that number. (e.g., 2 x 1 = 2; 4 x 1 = 4) Five Rule (skip count by fives) Any number multiplied by 5 means to count by fives that number of times (e.g., 2 x 5 means to count five twice –” five, ten;” 4 x 5 means to count five four times – “five, ten, fifteen, twenty.”) Nine RuleWhen multiplying a number by 9: subtract 1 from the multiplier to get the tens digit; count up to nine from that number to get the ones digit (e.g., 2 x 9 – one less than 2 is one so the tens digit in the answer is 1; then I count up from one to nine and I get eight so 8 is the ones digit in the answer. So, 2 x 9 = 18: 1 is one less than 2 and 1 + 8 = 9). Once students master these rules only 15 facts are left to multiply: 3x3=9; 3x4=12 3x6=18 3x7=21 3x8=24 4x4=16 4x7=28 4x8=32 6x6=36 6x7=42 6x8=48 7x7=49 7x8=56 8x8=64

8. 3. Create unique ways to remember less familiar facts with your students. FactExamples 8 x 7 = 56“5, 6, before 7, 8” 6 x 7 = 42Seven chops six into a 4 and a 2 8 x 9 = 72“Seven eight nine and left a two” 6 4 2 Selecting facts that are difficult for students and engaging with them to create ways to make facts meaningful can be an important activity for stimulating students’ metacognitive thinking and relating something abstract and meaningless to something tangible and meaningful.

9. 3. Computation - teach low stress alternative algorithms (adapted from Mercer & Mercer, 2001) AdditionPartial Sums 35 +27 12 60 72 Left-to-Right 788 +426 11 10 14 1214 SubtractionSubtract a Constant (to avoid regrouping) 500 – 1 499 -282 – 1 - 281 218 Add a Constant (to avoid regrouping) 37 + 1 38 -29 + 1 -30 8

10. 3. Computation - teach low stress alternative algorithms (adapted from Mercer & Mercer, 2001) MultiplicationPartial Products 35 X 7 35 210 245 Drop Notation 8 X 6 4 8 48 Division 11 __251_____ 21) 7581 42 33 21 128 105 23 21 0 34 X 3 12 90 102 Answer: 361 This algorithm helps students who have difficulty with the multiplication process - determining the correct quotient number when dividing. Initially student tried “2” but after subtracting 42 from 75 found that “33” was greater than the divisor. So divides by one more (“1” is placed above “2” in the quotient). Student repeats this step until no more multiplication is needed.

11. Fraction Number Sense – Number Line The following set of slides demonstrate how a number line and concrete materials (fraction bars) can be used to help students develop number sense related to fractions. The questions below each representation are those a teacher might ask. The words in the light colored boxes are language teachers and students might use to describe the meaning of each new representation.

12. Number Line Example: Fractions 0 1 How many blocks are there between zero and one on the number line? Is each block the same size? Let’s compare them to find out. How could I do that?

13. Number Line Example: Fractions 0 1 How many blocks are there between zero and one on the number line? How many blocks are on top of the number line? One of the four blocks is on top of the number line

14. Number Line Example: Fractions 0 1 How many blocks are there between zero and one on the number line? How many blocks are on top of the number line? I’m going to represent one of four blocks by writing one- fourth on the number line. 1/4

15. Number Line Example: Fractions 0 1 How many total blocks are there between zero and one on the number line? How many blocks are on top of the number line? Now there are two of four blocks on the top of the number line. 1/4

16. Number Line Example: Fractions 0 1 How many total blocks are there between zero and one on the number line? How many blocks are on top of the number line? I’m going to represent two of four blocks by writing two- fourths on the number line. 1/42/4

17. Number Line Example: Fractions 0 1 How many total blocks are there between zero and one on the number line? How many blocks are on top of the number line? Now there are three of four blocks on the top of the number line. 1/42/4

18. Number Line Example: Fractions 0 1 I’m going to represent three of four blocks by writing three-fourths on the number line. 1/42/4 3/4

19. Number Line Example: Fractions 0 1 How many total blocks are there between zero and one on the number line? How many blocks are on top of the number line? Now there are three of four blocks on the top of the number line. 1/42/4 3/4

20. Number Line Example: Fractions 0 1 Why do you think I wrote four-fourths underneath the number one on the number line? I’m going to represent three of four blocks by writing four-fourths on the number line. 1/42/4 3/4 4/4