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 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 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 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 over generalize that answers to addition are always greater. Doubles Many times students will use doubles as an anchor when adding and subtracting.
TASKS Task Name Standards Task Type/ Grouping Strategy Content Addressed Brief Description MGSEK.CC.2-4a, b, c 3-Act Task Whole Group Subitizing, Counting, Sequencing numbers Students use counting and sequence of numbers to figure out what comes next. Scaffolding Whole/Small/Partner/Individual Subitizing, Counting objects to 10, Sequencing Numbers, Students practice counting and subitizing objects to 10 in a variety of activities. MGSEK.CC.2, 4a,b,c Constructing Task Whole/Small/Partner/Individual Subitizing, Counting objects to 10, Sequencing Numbers, Matching Number Words to Numbers Students begin to make the connection between numeral, pictures and words in a variety of activities. MGSEK.CC.1-4 Constructing Task Whole/partner Numeral recognition, number word recognition, Numeral writing Students practice connecting numerals to sets. MGSEK.CC.1-4 MGSEK.MD.3 Constructing Task Partner Subitizing, Modeling numbers, Understanding number relationships Students make a Rekenrek. Students also practice modeling numbers through a variety of activities. MGSEK.CC.2, 4 Practice Task Whole/Small/Partner/Individual One to one correspondence Students practice counting and making a set of objects to represent a number.
MGSEK.CC.1, 2, 4 Practice Task Partner One to one correspondence Students use one to one correspondence to count their way around a game board. MGSEK.CC.1-4 FAL One to one correspondence, Counting Objects to 10, Numeral recognition, Understanding number relationships Students are assessed on the standards taught so far in the unit. This allows teachers to address any students who may need remediation or acceleration. MGSEK.CC.1, 2, 4 Constructing Task Whole/Partner One to one correspondence, Counting objects to 10, Numeral recognition Students practice counting forwards and backwards as they participate in a variety of activities. MGSEK.CC.1, 2, 4MGSEK.MD.3 Constructing Task Partner Sequencing Numerals, numeral recognition Students practice ordering sets from least to greatest. Constructing Task Whole/Partner Numeral recognition, Understanding number relationships Students begin to understand more and less in a variety of activities. MGSEK.CC.1-4 MGSEK.MD.3 Constructing Task Small groups, individual Numeral recognition, Understanding number relationships Students practice connecting sets to numerals and words. MGSEK.CC.1, 2, 4b,c Practice Task Whole/Partner Numeral recognition, Sequencing Numerals, Understanding number relationships Students practice their understanding of more or less to guess their partners number on a number line.
What is the number of the next space? How can playing board games make me a better mathematician? DIFFERENTIATION Extension Each time a student rolls a die they alternate between counting spaces forwards and backwards. The number of spaces is determined by the amount shown on the die. Example: If player one rolled a six, they would move forward six spaces. If on the next turn, player one rolled a three, they would move backwards three spaces. Students would continue on until a player reached 20. Intervention Because the students must say the numerals aloud, the numerals could also be written on the game board to assist with location identification and sequential counting. Back to Intervention Table TECHNOLOGY Okta’sRescue: http://illuminations.nctm.org/ActivityDetail.aspx?ID=219
CONSTRUCTING TASK: Order the Dice Approximately 1 day STANDARDS FOR MATHEMATICAL CONTENT MGSEK.CC.1 Count to 100 by ones and by tens. MGSEK.CC.2. Count forward beginning from a given number within the known sequence (instead of having to begin at 1). MGSEK.CC.4 Understand the relationship between numbers and quantities; connect counting to cardinality. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. (one-to-one correspondence) Understand that the last number name said tells the number of objects counted (cardinality). The number of objects is the same regardless of their arrangement or the order in which they were counted. Understand that each successive number name refers to a quantity that is one larger. MGSEK.MD.3.Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. (For descriptors of standard cluster please see the Grade Level Overview) 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. 6. Attend to precision. 7. Look for and make use of structure.
Part II (More/Less 1&2): Example: if you roll a 5 and spin 2 more, you count forward 2 from 5 to end at seven. As students play, they record the number they rolled on the recording sheet. Then they record what they spun (more/less, 1 more, 1 less, etc….). Students then record what they covered on the game board. They justify this in the “because” section by writing an equation or another justification for covering. (Example: A player could say she rolled one more than 8. That’s 9, because one more is the next number, so in the space she wrote “it’s next.”) First player to get 3 counters in a row wins. (SMP 1-8) TEACHER REFLECTION QUESTIONS Are students able to identify numbers 0-10? Are students able to identify 1 more or less than a given number? Are students able to identify 2 more or less than a given number? Are students abe to write numerals correctly? FORMATIVE ASSESSMENT QUESTIONS How do you know that you counted correctly? What does “more” mean? What does “less” mean? What numbers do you need to win? Why did you choose that number? If you spun “2 more” what number would you need to roll to win? DIFFERENTIATION Extension Have the students model their actions using a ten-frame or Rekenrek. This will also help students to record their actions. Intervention Allow the students to model with a ten frame or through the use of a number line. Back to Intervention Table (0-8) More or Less -3 in a Row Materials: spinner dice Rules: Player 1 rolls the dice and spins the spinner. Player 1 covers a number space that relates to the spinner. (Example: if player 1 rolls a 5 and spins “less”, they can cover any number less