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Place Value

  1. The following is a set of interview data (using diagnostic interview tasks) taken over a week in the early part of the year with a class of second grade students. The interviews went as follows:
    a. “Please write the number sixty-seven.” (All can do this.) “Now write the number that is one more than that number.” (Most can do this, also.) “Now write the number that is 10 more than 67.” In the sample, all of the students counted using their fingers and only a few were successful. Not a single child was able to write the number that was ten less. All tried to do so by counting backwards.
    b. The digit correspondence task was done using 53 counters exactly as described in the text on page 220. There were no students in the group that evidenced a place-value understanding of the 5 in 53.
    c. Students were shown a clear plastic baggy of 57 small counters and asked, “About how many do you think are in the bag?” Students were fairly successful at making reasonable estimates. Note that this first aspect of the task requires no place value understanding. Then the students were asked to help count out the contents by putting the counters into groups of ten. That was followed with beginning the “Fill the Tens” task (shown in Figure 10.5 p 219 in the text). The teacher then filled one ten-frame card and began the second. “If we keep on filling up these cards like this, how many cards will we need before we run out of these 57 counters?” Not one child got this correct. Many said we would need 57 cards. Most students simply had no idea. In the classroom, these same students were able to write the number for a tens and ones picture of rods and cubes, could read and write numbers, and could find numbers easily on the hundreds board.
    What can you learn from the students’ responses to these questions?
    What would the next steps be for these students?
  2. Find the number given the clues below
    • Less than half of 100
    • The product of the digits is less than one dozen
    • The sum of the digits is more than 6
    • Not a multiple of 5
    • Containing only digits less than 9
    • Containing only odd digits
    • The tens place is less than the number of sides on a pentagon
    • An odd number
    • A number that does not contain the smallest prime odd number as a digit
    • Between one dozen and two dozen

What is the number? __

Addition & Subtraction

  1. Solve these without using the standard algorithm. Describe the method you used for each (e.g., empty number line, splitting, shortcut, bar diagram, partial sums, etc).

465 + 230

526 + 98

7000 – 25

342 + 153 + 481

Multiplication & Division

  1. Solve these without using the standard algorithm. Describe the method you used for each (e.g., complete-number strategies, partitioning, compensation, cluster, etc).

35 x 12

86 x 42

45 x 6

  1. Solve these without using the standard algorithm. Describe the method you used for each (e.g., missing-factor, cluster, decomposition, etc).

1224 ÷ 24

583 ÷ 4

  1. Would you be learning more about addition, subtraction, multiplication, division concepts if you had used the standard algorithm—or by using your approach?

Sample Solution

making support available earlier to those who do not yet require personal care, but may benefit from care counselling and information and home adaptations. 4. Integration between health and care People who use social care often use a number of other services, particularly community and acute healthcare. The NHS and local authorities who purchase and provide these services share a number of goals: keeping people well in their own homes for as long as possible, improving system efficacy and promoting a more person (or patient) centred approach. However, the experience of using a both services can often be fragmented, frustrating and sometimes detrimental to wellbeing. For instance, people often find themselves in hospital unnecessarily due to insufficient coordination between hospital discharge and local authority assessment processes. Integrating these services to provide a more coordinated service has been a key policy goal for UK governments for at least two decades. There have been a number of different approaches to this challenge. These approaches include pooled funding, joint planning and purchasing of services as well as merging organisations, co-locating staff and centralising information systems. Miller et al (2016) describe a number of ways of thinking about integration, including the micro, meso and macro spectrum. Micro integration describes the interactions of practitioners, for instance domiciliary care workers and district nurses. Meso integration describes joint teams of practitioners from different silos or targeted services such as integrated discharge teams. Macro integration describes the systems-level integration at play in joint boards and shared policymaking across localities. Across the UK, recent integration policy has focused on macro-level changes: systems level funding and strategy for particular localities, with meso and micro initiatives greatly varying from area to area.
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