Hopping Backward to Solve Problems
Unit: How Many More Fish?
PreK to 2nd
Grace M. Burton
Using chalk or masking tape, make a number line on the floor. (The
students will use this to find differences on a number line by hopping
from a number toward 0.) Tell the students that they will now use the
number line to compare lengths. Ask one student to hop to 5 and another
to hop to 3. Then ask, “Who hopped farther? How much farther?” Repeat
with other students.
Next, draw a number line with the spaces one cracker apart, draw a
red ring and place 3 fish-shaped crackers and a blue ring with 2
fish-shaped crackers inside. Ask: How many more fish-shaped crackers
are in the ring with 3 fish-shaped crackers? How can we find out using
the number line?
Encourage the students to line up the crackers from the red ring with the left end of the number line.
Then ask them to place the crackers from the blue ring in a line below the first line.
Next show how to hop back from the end of the longer line, counting
the hops aloud. Have the students record the comparison using the
equation notation [3 - 2 = 1] on the Differences Activity Sheet.
Differences Activity Sheet
It is not uncommon for the students to count the lines on the
number line rather than the spaces covered by the hops. You may wish to
highlight the fact that in this meaning for the operation of
subtraction, spaces are counted, not points on the number line. You may
demonstrate this by using a length of paper the size of a fish-shaped
cracker to hop back with. After several examples, show the students
that they do not need to place the crackers themselves on the number
line, but can mark the length with a crayon.
To enrich the students' understanding of the number line concept, model how to use the Number Line Arithmetic Tool
from the National Library of Virtual Manipulatives to compare lengths.
Encourage the students to use this site during math center time, and
assign students to work at the site in pairs. Those not taking their
turn at the computer should complete the next activities.
Put the students into pairs and give each pair fish-shaped crackers, crayons, and one number line from the Number Lines Activity Sheet.
Number Lines Activity Sheet
Ask each student to make two sets of crackers on a piece of
paper, and then enclose each in rings of different colors. Then have
the students line up the crackers carefully and draw, in the
appropriate colors, a line as long as the number of crackers in the
set. Then ask them to compare the lengths on the number line to find
the difference and to record the comparison in pictures and in equation
form. After allowing time for exploration, call the students together
to read their equations and share their number line illustrations.
As a concluding activity, pose puzzles such as "I am thinking of two
numbers on the number line that have a difference of 5. The larger
number is 6. What is the other number?" (If the students are ready for
a challenge, you might say only: "I am thinking of two numbers on the
number line that have a difference of 5. What are the numbers?") You
may wish to have the students create and share similar problems. One or
more of these puzzles could be added to their learning portfolios.
You may find it helpful to add to your recordings on the Class Notes recording sheet you began earlier in this unit. This data may be helpful as you plan strategies for regrouping students.
Questions for Students
1. How could you use the number line to compare two plates, one of which has five fish-shaped crackers and the other of which has three fish-shaped crackers?
[Draw a number line, mark the places for five and three with fish, and then compare the distance between them.]
2. What numbers have a difference of 2? Can you find some of them on the number line?
[Some examples include: 5 - 3, 4 - 2, and 3 - 1.]
3. What would be the difference if two plates had the same number of crackers on them? Would the lines that showed how many crackers are in each plate be the same length? How do you know?
[0; yes; student responses may vary.]
4. How would you explain to a friend how to compare lengths on the number line?
[Student responses may vary.]
Pre K to 2nd
Grade: PreK to 2nd
How Many More?
Fact Family Fun
Wrapping Up the Unit