How many more, or how many times?
Compare the same quantities by difference and multiplier, naming the reference.
Begin with the activity ↓How many more, or how many times?
Compare 2 beads and 6 beads. Six is 4 more than two, and 3 times two. Subtraction finds the difference; division finds the multiplier.
Compare 2 beads and 6 beads. Six is 4 more than two, and 3 times two. Subtraction finds the difference; division finds the multiplier.
The same two groups support different questions. “How many more?” asks for the extra count. “How many times?” asks how many reference groups fit. Decide what the answer means before choosing a calculation.
Treat two beads as one reference group. Six makes three such groups. In 6 ÷ 2 = 3, two is the reference. Three is a multiplier, not a bead count. Say the full statement: six beads is three times two beads.
Changing the reference changes the multiplier. With six as the reference, 2 ÷ 6 = 1/3, so two is one third of six. A multiplier need not exceed one.
Align units when comparing lengths. One metre and 50 cm become 100 cm and 50 cm: a multiplier of two. Dividing the bare numbers 1 and 50 loses the units. A zero reference would require division by zero, so we exclude it.
Keep the reference fixed and change the compared group below. Equal differences need not give equal multipliers. Next we use this distinction to write two quantities in an agreed order as a ratio.
Listen one paragraph at a time
Hold on to this distinction
Two and six differ by four with multiplier three. Four and eight also differ by four, but the multiplier is two. Say which quantity you divided by.
Change it and check
Change both counts and compare the two bars. Predict the difference and quotient before checking.
Compared − reference: 6 − 2 = 4 items
Compared ÷ reference: 6 ÷ 2 = 3 times
When the compared count is smaller, the subtraction is negative. The size of the difference is 4. Review negatives
The reference is a count from 1 to 8; the compared count is 0 to 12. Each bar cell represents one item, with matching units.
Try with paper and objects
Lay out 2 and 6 bottle caps. Mark the four extra caps, then group the six into three groups of two.
Check and explain another example
Check your idea 1
Check your idea 2
Apply it in a new setting
Write an answer yourself
Return to the explanationCompare with a solution
4 · times
Four groups of three fit: 12 ÷ 3 = 4.
Does your explanation include these ideas?
Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.
Continue, or pause here
Name ratio order and reference, distinguishing part-to-part from part-to-whole.