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 Tells 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 learn 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 Using 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 greater. Doubles Many times students will use doubles as an anchor when adding and subtracting.
COMMON MISCONCEPTIONS Students may over-generalize the vocabulary in word problems and think that certain words indicate solution strategies that must be used to find an answer. They might think that the word more always means to add and the words take away or left always means to subtract. When students use the words take away to refer to subtraction and its symbol, teachers need to repeat students’ ideas using the words minus or subtract. For example, students use addition to solve this Take from/Start Unknown problem: Seth took the 8 stickers he no longer wanted and gave them to Anna. Now Seth has 11 stickers left. How many stickers did Seth have to begin with? If students progress from working with manipulatives to writing numerical expressions and equations, they skip using pictorial thinking. Students will then be more likely to use finger counting and rote memorization for work with addition and subtraction. Counting forward builds to the concept of addition while counting back leads to the concept of subtraction. However, counting is an inefficient strategy. Teachers need to provide instructional experiences so that students progress from the concrete level, to the pictorial level, then to the abstract level when learning mathematics. TASKS The following tasks represent the level of depth, rigor, and complexity expected of all Kindergarteners. These tasks or a task of similar depth and rigor should be used to demonstrate evidence of learning. TASK DESCRIPTIONS Scaffolding Task Tasks that build up to the learning task. Constructing Task Constructing understanding through deep/rich contextualized problem solving tasks. Practice Task Tasks that provide students opportunities to practice skills and concepts. Culminating Task Designed to require students to use several concepts learned during the unit to answer a new or unique situation. Allows students to give evidence of their own understanding toward the mastery of the standard and requires them to extend their chain of mathematical reasoning. Formative Assessment Lesson (FAL) Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key mathematical ideas and applications. These lessons enable teachers and students to monitor in more detail their progress towards the targets of the standards.
Task Name Standards Task Type/ Grouping Strategy Content Addressed Brief Description MGSEK.OA.1-5 Constructing Task Individual. Whole, Small Group Addition and subtraction through word problems Students answer different type addition and subtraction story problems MGSEK.OA.1,3,4,5 Constructing Task Whole or Small Group Number combinations to 5 and 10 Students practice different ways of building five using two different quantities MGSEK.OA.1-5 Practice Task Whole, Small Group or Individual Number combinations to 5 through problem solving Students find different combinations to build 5 and 10 MGSEK.OA.1,4 Practice Task Partners Using a number line to find number combinations to 20 Students practice adding and subtracting to 10 MGSEK.OA.1,5 Practice Task Partners Number combinations to 5 (+/-) Students practice finding the benchmark number 5 when adding to 20 Shake and Spill Addition/Subtraction MGSEK.OA.1,3,4,5 Constructing Task Whole, Small Group or Individual Number combinations to 10 (+/-) Students build number combinations to 10 MGSEK.OA.1,3,4,5 Practice Task Partners Using 5 as a benchmark number, number combinations to 10 Students practice using the benchmark number 5 to build number combinations to 10 MGSEK.OA.1-5 FAL Using 5 and 10 as benchmarks Addition and Subtraction Students show their understanding of benchmark numbers in a formative assessment lesson
MGSEK.OA.1-5 Practice Task Whole, Small Group or Individual Number combinations to 5 and 10 Students use a variety of activities to build combinations to 10 MGSEK.OA.1-5 Performance Task Individual Number combinations to 5 through problem solving Students use different combinations to five to solve a problem. MGSEK.OA.1-5 Constructing Task Whole, Small Group or Individual Mental computation of number combinations to 5 or 10. Students practice building number combinations to 10 mentally MGSEK.OA.1-5 Performance Task Individual Number combinations to 5 through problem solving Students practice building number combinations to 5 MGSEK.OA.1-5 Practice Task Partners Mental computation of number combinations to 5 or 10. Students practice building number combinations to 5 and 10 MGSEK.OA.1-5 Practice Task Whole, Small Group or Individual Number combinations to 5 through problem solving Students practice building number combinations to 5 MGSEK.OA.1-5 Practice Task Partners Mental computation of number combinations to 5 or 10. Students practice building number combinations to 5 and 10 MGSEK.OA.1-5 Culminating Task Individual Number combinations to 5 through problem solving Students use their knowledge gained to build number combinations to 5 Each task is suggested but not required; teachers should choose the most appropriate tasks based on the needs of their students. For more information on these tasks in this unit please refer to the unit webinar found at
Compare3 (“How many more?” version): Spiders have 8 legs. Mice have 4 legs. How many more legs do spiders have than mice? (“How many fewer?” version): Spiders have 8 legs. Mice have 4 legs. How many fewer legs do mice have than spiders? (Version with “more”): Mice have 4 legs. Spiders have 4 more legs than mice. How many legs do spiders have? (Version with “fewer”): Mice have 4 fewer legs than spiders. Spiders have 8 legs. How many legs do mice have? (Version with “more”): Spiders have 8 legs. They have 4 more legs than mice. How many legs do mice have? (Version with “fewer”): Mice have 4 fewer legs than spiders. Spiders have 8 legs. How many legs do mice have? The tasks in this chart provide a variety of word problems using addition and subtraction for students to explore. It is not necessary for students to be able to solve ALL problem types in order to meet the standard for solving word problems using objects or drawings (MGSEK.OA.2) CONSTRUCTING TASK: Building Trains (Addition/Combining) Approximately 1 day and repeated through centers (adapted from Chris Confer’s “Snap It” found in Teaching Number Sense) STANDARDS FOR MATHEMATICAL CONTENT MGSEK.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings1, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. MGSEK.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. (drawings need not include an equation). MGSEK.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. MGSEK.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.
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 - = - =
Calling Out Reds? Name:_________________________ Number of Red Cards I have + How many red cards my partner has = Total Number of Red cards PRACTICE TASK: Dropping Pennies Back To Task Table Approximately 1-2 days
The Candy Store Emma and Audrey went to the candy store and they had 2 nickels to spend between them. If they spent both nickels and each got only 1 item from the candy store, what are some of the items they could have bought? Free Apple 1 cent sucker 2 cents lollipop 3 cents gum 4 cents Chocolate bar 5 cents milkshake 6 cents chips 7 cents pop 8 cents jellybeans 9 cents popcorn 10 cents Hotdog & pop
-------------------------------------------------------------cut along line---------------------------------------------------------------------------------