1. Buddhist Monk Puzzle One morning, exactly at sunrise, a Buddhist monk leaves his temple and begins to climb a tall mountain. The narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit. The monk ascended the path at varying rates of speed, stopping many times along the way to rest and eat the dried fruit he carried with him. He reached the temple shortly before sunset. After several days of fasting, he begins his journey back along the same path, starting at sunrise and again walking at variable speeds with many pauses along the way, finally arriving at the lower temple just before sunset. Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.

2. Moving From Counting to Addition and Subtraction “too often teachers move directly from the beginning ideas of counting to addition and subtraction, leaving children with a very limited collection of ideas about number to bring to these new topics” (Van de Walle, 2014, p. 107).

3. Relationships between Numbers 1 through 10 Spatial relationships Dot plates Dot-pattern dominoes One more, one less, two more, two less Anchor to 5 and anchor to 10 Five-frame Ten-frame Part-part-whole Unifix cubes Two-color counters Dot strips

4. Word Problems “when kindergarten and first-grade children are regularly asked to solve word problems, not only do they develop a collection of number relationships, but they also learn addition and subtraction facts based on these relationships. The key is to allow them to figure out ways to solve the problems” (Van de Walle, 2014, p. 116).

5. Understanding 10 Initial conception of 10 10 is ten 1s Do not see 10 as a unit Intermediate concept of 10 Sees 10 as a unit Needs manipulatives or other representations to work with 10s Facile concept of 10 Thinks of two-digit numbers as groups of 10s and 1s Do not need representations to work with 10s

6. Teen Number Names Not “place-value” names, e.g., “eleven” vs. “ten-one” Confusing (lack of) pattern eleven, twelve thirteen, fifteen fourteen, sixteen, seventeen, eighteen, nineteen 20-99 also not “place-value” names, e.g., “fifty-three” vs. “5-tens- and-three”

8. Real World Problems Calendar activities = “add on” activities Estimation and measurement Graphs Calendar activities = “add on” activities Estimation and measurement Graphs

9. Process Standards https://www.teachingchannel.org/videos/visualizing-number-combinations

10. Which Is More? A can of tennis balls holds three balls packed tightly against the top, sides and bottom. Which is more, the height of the can or the distance around the can?