Number Sense Trajectory –Putting It All Together Trajectory Subitizing Being able to visually recognize a quantity of 5 or less. Comparison Being able to compare quantities by identifying which has more and which has less. Counting Rote procedure of counting. The meaning attached to counting is developed through one-to-one correspondence. One-to-One Correspondence Students can connect one number with one object and then count them with understanding. Cardinality how many things are in a set. When counting a set of objects, the last word in the counting sequence names the quantity for that set. Hierarchical Inclusion Understands numbers are nested inside of each other and that the number grows by one each count. 9 is inside 10 or 10 is the same as 9 + 1. Number Conservation The total number of objects remains the same when they are rearranged spatially. 5 is 4&1 OR 3&2. Each concept builds on the previous idea and students should explore and construct concepts in such a sequence Number Relationships Spatial Relationship Patterned Set Recognition Students can to recognize sets of objects in patterned arrangements and tell how many without counting. One and Two-More or Less Students need to understand the relationship of number as it relates to +/- one or two. Here students should begin to see that 5 is 1 more than 4 and that it is also 2 less than 7. Understanding Anchors Students need to see the relationship between numbers and how they relate to 5s and 10s. 3 is 2 away from 5 and 7 away from 10. Part-Part-Whole Relationship Students begin to conceptualize a number as being made up from two or more parts. Addition and Subtraction Strategies One/Two More/Less These facts are a direct application of the One/Two More/ Less than relationships Make a Ten Use a quantity from one addend to give to another to make a ten then add the remainder.9 + 7 = 10 + 6 Near Doubles the doubles anchor and combining it with 1 and 2 more/less. Facts with Zero Need to be introduced so that students don’t overgeneralize that answers to addition are always bigger. Doubles Many times students will use doubles as an anchor when adding and subtracting.
Building Trains (Addition) Name:__________________________________ Is the same as + = and is equal to + and Is the same as + = + =
Building Trains (Subtraction) Name:__________________________________ Is the same as - = and is equal to - minus Is the same as - = - =
CONSTRUCTING TASK: Shake and Spill Addition (5 Or 10) Approximately one day and repeated through centers (adapted from Marilyn Burns’ Shake and Spill activity found in About Teaching Mathematics: A K-8 Resource STANDARDS FOR MATHEMATICAL CONTENT MCCK.OA.1. Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. MCCK.OA.3.Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). MCCK.OA.4.For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation. MCCK.OA.5.Fluently add and subtract within 5 STANDARDS FOR MATHEMATICAL PRACTICE 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. BACKGROUND KNOWLEDGE “When parts of a set are known, addition is used to name the whole in terms of the parts.” (Van de Walle & Lovin 2006) Shake and Spill allows children to focus on a single number for the entire activity. It is important to give the children time to work on a single number (usually to 4 or 5, at first) throughout the activity, allowing them opportunities to explore through a variety of materials and methods of joining or separating. As their understanding of concepts develop, encourage students to extend their understanding with higher numbers. Allowing students multiple opportunities to participate in these types of activities gives them the chance to think about number relationships in a relaxed setting (Burns 2007; Van de Walle & Lovin 2006).
Intervention Students may use a part-part-whole mat for this activity. Make sure that students flip the mat so that the “whole” section is first, showing that the one part is removed from the whole. An example of a part part whole mat can be seen below. Whole part part TECHNOLOGY Math Games: Popup Subtraction Shake and Spill: Subtraction Total Number of counters - Number of Yellow Counters = is the same as: Number of Red Counters
Calling Out Reds? Name:_________________________ Number of Red Cards I have Number of Red Cards I predict my partner has How many red cards my partner actually has Total Number of Red cards