New Zealand Curriculum: Level 4
Learning Progression Frameworks: Measurement sense, Signpost 7 to Signpost 8
Target students
These activities are intended for students who use a range of strategies for the addition and subtraction of whole numbers. These strategies may include elements of integer thinking such as solving 52 – 28 by first solving 52 – 30, then compensating by adding two, i.e., 52 – 28 = 52 – 30 + 2 = 24. Students should also know their basic addition facts and the corresponding subtraction facts.
The following diagnostic questions indicate students’ understanding of, and ability to locate integers on a number line, and to add and subtract both positive and negative integers. Allow access to pencils and paper. A calculator can be used to check answers if needed. (show diagnostic questions)
The questions should be presented orally and in a written form so that the student can refer to them. The questions have been posed using a money context but can be changed to other contexts that are engaging to your students.
[Use Copymaster page 2]
Here is a number line from positive ten to negative ten. Point to where these numbers would be on the line.
- Zero (0)
- Five (5)
- Negative Five (-5)
- Negative Eight (-8)
I will point to a place on the number line. Write the number that would be in that place.
[Point to the location of 4, -2, and -11]
Signs of fluency and understanding:
Recognises the symmetry of the integer number line about zero to locate zero, positive five, and negative five. Moves three imaged units left from -5 to locate -8. Records the correct numbers for the three locations.
What to notice if they don’t solve the problem fluently:
Confuses the direction of negative numbers compared to positive numbers. May record -8 to the left of -10. This indicates that the student needs more experience with creating and reading integer number lines, such as working with thermometers and scales above and below sea level.
Unable to equally divide sections of the number line to locate fractions. May point to a place close to -10 for -8 without considering the size of each unit. This indicates that the student needs more experience with creating measurement scales by equal partitioning spaces.
Unable to record negative numbers. This suggests that the student needs more exposure to integer scales.
Supporting activity:
Integer number lines
- [Use Copymaster page 3]
A circle with a positive sign means positive one. A circle with a minus sign means negative one.
Match the pictures with the numbers.
Explain what you have done.
Signs of fluency and understanding:
Understand that positive one and negative one combine to make zero. Recognise that three negatives and one positive combine to make negative two, and that positive two and negative two combine to make zero.
What to notice if they don’t solve the problem fluently:
Confused by the combination of positive and negatives. Unsure of how to combine three negatives and one positive. This indicates that the student needs more experience with applying integers to real world situations like dollars and debts.
Unable to represent any of the collections using integers. This indicates that the student does not recognise that integers are counts of positive and negative ones.
Supporting activity:
Representing quantities as integers
- [Use Copymaster page 4]
Poppy runs a bakery stall to raise money for her sports trip. She collects $450 in sales and her costs are $550. How much money does her stall make?
Which equations represent this story?
Signs of fluency and understanding:
Easily finds the answer as -$100 (negative or minus one hundred dollars).
Recognises that both 450 + -550 = -100 and 450 – 550 = -100 represent the situation.
What to notice if they don’t solve the problem fluently:
Selects 550 – 450 = 100 as the equation. This may indicate the student believes that subtraction must always involves a larger number minus a smaller number. They may believe that “subtraction makes smaller" and that "subtraction can't go below zero.”
Selects 450 – 550 = -100 as the only equation that matches the situation. This may indicate that the student does not recognise that combining positive and negative amounts can be represented as addition.
Supporting activity:
Adding integers
- [Use Copymaster page 5]
Which number line represents -5 - -3 = [ ]?
Signs of fluency and understanding:
Recognises that subtracting negative three is equivalent to adding positive three. Chooses the top right number line and correctly give the answer as negative two (-2).
What to notice if they don’t solve the problem fluently.
Selects the top left number line and gives the answer negative eight. This may indicate that the student believes that subtraction always results in an answer that is smaller than the starting number.
Selects the bottom left number line. The student may be simply choosing an option that contains 5 and 3. It may also indicate confusion between "–" as the symbol for subtraction and "–" as the symbol for direction on the number line.
Selects the bottom right number line. This may indicate that the student believes that -3 + 5 has the same answer as -5 - -3. The student needs support to understand that the order of the numbers matters for subtraction.
Supporting activity:
Subtracting integers
Teaching activities
