Academics · Elementary · Grades 2 to 4

Build the fact.
Then it is yours.

Tap a board of counters to make rows and columns. The page writes the times fact, breaks a hard one into two easy ones and shows the same counters as a division. Then practice, with a way in after every miss.

Open the array boardWhat you get
78facts to learn, up to 12 × 12
12worked strategies, one per table
24array hunts in three grade bands

The array lab

Tap or drag on the board to set the rows and columns. Pick the grade first: it sets the size of the board and what the page says about each array.

Loading the array board. If nothing appears, turn on JavaScript.

Every hard fact has an easy one inside it

Practice gives one of these hints after a miss, with the picture. Each is drawn here for the sevens.

× 2

Double it

2 × 7 is 7 + 7 = 14.

× 4

Double, then double again

4 × 7: double 7 is 14, double that is 28.

× 8

Double three times

8 × 7: 14, 28, 56.

× 5

Half of ten times

Ten 7s make 70. Half of that is 35.

× 9

Ten times, take one away

Ten 7s make 70. Take one 7 away: 63.

× 6 and × 7

Start from five

7 × 7: five 7s make 35, two more make 14. Together 49.

Ones and tens need no trick, and × 3 is a double plus one more group. That covers every fact to 12 × 12. A fact that is worked out this way a few times usually ends up remembered, and it can always be worked out again.

What you get

Academics · ElementaryFree

What you finish with
Facts built by hand and seen four ways (rows, a turned array, two easier facts, a division), up to eight array hunts per grade, practice rounds with a worked strategy for every missed fact, and a facts table that shows which facts are known. A printed chart and practice sheets go in the folder.
Who it is for
Students in grades 2 to 4 learning what multiplication and division mean and building the facts; older students who never got the facts down; parents, homeschool families, tutors and teachers who want an array station with no counters to pick up afterward.
Format
Browser lab with four parts: Arrays, Sharing, Practice and My facts. Works by touch, mouse or keyboard on a phone, tablet or computer. Prints a facts chart and a 24-question practice sheet with an answer key; saves the facts list as a .csv.
Access
Free. No account and no sign-in. The board, the hunts and the practiced facts are saved only in this browser; no names are asked for and nothing is sent to Caveat.
Where it lives
The Elementary space in Caveat Academics.
Support
Questions, a correction or a request: team@caveatacademic.com.
The Sharing tab deals counters into groups and keeps the leftovers in view, so a remainder is something a child can see.

How to use it

  1. Pick the grade, then tap the board. The spot you tap becomes the bottom right corner of the array.
  2. Read the dark strip and the rows under it: count by rows, turn it, and slide Break apart to split a hard fact.
  3. Try the array hunt beside the board, then open Sharing to see the same numbers as a division.
  4. Play a round in Practice. My facts fills in as you go; print the chart or a practice sheet when you want paper.

Open the array board

How the lab stays right

Everything on the page is computed from the counters on the board, and every hint is checked before the page is built.

What the lab works out

  • The fact and its family come straight from the rows and columns on the board, so the picture and the numbers always agree.
  • Breaking apart splits the columns at the slider: 6 × 7 becomes (6 × 5) + (6 × 2). This is the distributive property, shown before it is named.
  • The hints in Practice use one rule per table (double, double twice, half of ten, ten take one away, five and some more). The build works each hint out the way it is worded for all 144 facts and stops if any hint misses its product.
  • Known means right on the first try twice in a row. A miss resets it. There is no timer and no score for speed.
  • Hunts count 3 × 8 and 8 × 3 as one array, and name the pairs that are too long for the board.

For the adult nearby

Start on Arrays, not Practice. Ask for a prediction before the board is tapped (“how many will 4 rows of 6 be?”), then let the count-by row settle it. In Grade 2 the lab stays with rows-of language and adding, which is what second graders are asked to do.

When a fact is missed, read the hint aloud together and build the same fact on the board with the slider. One hard fact a day, met three ways, sticks better than a page of drill.

Sharing shows why 23 ÷ 4 has two stories: four friends sharing 23 cards, or 23 cards dealt in hands of four. Children need both. Real counters, coins or dried beans work just as well as the screen.

The three grade bands

BandBoardWhat it showsStandards it follows
Grade 25 by 5Rows-of sentences, adding equal rows, counting by rows, even and odd. No × sign yet.2.OA.C.3, 2.OA.C.4
Grade 310 by 10The × and ÷ signs, turning an array, breaking it apart, fact families, sharing two ways, facts to 10 × 10.3.OA.A.1–4, 3.OA.B.5–6, 3.OA.C.7
Grade 412 by 12Everything in Grade 3 to 12 × 12, plus factor pairs, prime and composite, times-as-many and remainders.4.OA.A.1–3, 4.OA.B.4

Sources

  1. Common Core State Standards for Mathematics (2010), Operations and Algebraic Thinking: odd and even by pairing, and rectangular arrays with up to 5 rows and 5 columns written as a sum of equal addends (2.OA.C.3–4); products as equal groups, quotients as shares or as groups, and unknown-factor problems (3.OA.A.1–4); properties of multiplication and division as an unknown-factor problem (3.OA.B.5–6); fluency within 100 and knowing the products of one-digit numbers from memory by the end of grade 3 (3.OA.C.7); times-as-many comparisons and interpreting remainders (4.OA.A.1–3); factor pairs for 1 to 100, prime and composite (4.OA.B.4).
  2. Common Core Standards Writing Team, Progressions for the Common Core State Standards in Mathematics: K, Counting and Cardinality; K–5, Operations and Algebraic Thinking (2011), on equal-groups, array and compare situations and on the path from counting all, to skip counting, to facts derived from known facts.
  3. The strategies and hunts are checked by an exact reference written for this page (guide/suite/mdf_ref.py): each of the 144 hints is worked out the way it is worded and compared with the product.

The lab covers the meaning of multiplication and division and the facts to 12 × 12. It does not teach multi-digit multiplication or long division. Practice with the ideas, not a full math course. Support: team@caveatacademic.com.